diff --git "a/39E4T4oBgHgl3EQfAwsS/content/tmp_files/2301.04845v1.pdf.txt" "b/39E4T4oBgHgl3EQfAwsS/content/tmp_files/2301.04845v1.pdf.txt"
new file mode 100644--- /dev/null
+++ "b/39E4T4oBgHgl3EQfAwsS/content/tmp_files/2301.04845v1.pdf.txt"
@@ -0,0 +1,5345 @@
+A groupoid approach to regular ∗-semigroups
+James East∗ and P.A. Azeef Muhammed
+Centre for Research in Mathematics and Data Science,
+Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia.
+J.East@WesternSydney.edu.au, A.ParayilAjmal@WesternSydney.edu.au
+Abstract
+In this paper we develop a new groupoid-based structure theory for the class of regular ∗-
+semigroups. This class occupies something of a ‘sweet spot’ between the important classes of
+inverse and regular semigroups, and contains many important and natural examples. Some
+of the most significant families include the partition, Brauer and Temperley-Lieb monoids,
+among other diagram monoids.
+Our main result shows that the category of regular ∗-semigroups is isomorphic to the
+category of so-called ‘chained projection groupoids’. Such a groupoid is in fact a pair (G, ε),
+where:
+• G is an ordered groupoid, whose object set P is a projection algebra (in the sense of
+Imaoka and Jones), and
+• ε : C → G is a special functor, where C is a certain natural ‘chain groupoid’ constructed
+from P.
+Roughly speaking:
+the groupoid G remembers only the ‘easy’ products in a regular ∗-
+semigroup S; the projection algebra P remembers only the ‘conjugation action’ of the projec-
+tions of S; and the functor ε tells us how G and P ‘fit together’ in order to recover the entire
+structure of S. In this way, our main result contains the first completely general structure
+theorem for regular ∗-semigroups.
+Among other applications, we use our structure theorem to obtain new, and perhaps
+more transparent, constructions of fundamental regular ∗-semigroups. We also use the chain
+groupoids to demonstrate the existence of free (idempotent-generated) regular ∗-semigroups
+associated to arbitrary projection algebras. Specialising to inverse semigroups, we also obtain
+a new proof of the celebrated Ehresmann–Schein–Nambooripad Theorem.
+We consider several examples along the way, and pose a number of problems that we
+believe are worthy of further attention.
+Keywords: Regular ∗-semigroups; inverse semigroups; groupoids; projection algebras; parti-
+tion monoids.
+MSC: 20M10, 20M50, 18B40, 20M17, 20M20, 20M05.
+Contents
+1
+Introduction
+3
+2
+Preliminaries
+9
+2.1
+Semigroups
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+2.2
+Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+2.3
+Regular ∗-semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+2.4
+Case study: diagram monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+∗Supported by ARC Future Fellowship FT190100632.
+1
+arXiv:2301.04845v1  [math.RA]  12 Jan 2023
+
+I
+Structure
+30
+3
+Projection algebras
+30
+3.1
+Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+31
+3.2
+The path category
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+34
+3.3
+Linked pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+38
+3.4
+The chain groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+40
+3.5
+The category of projection algebras . . . . . . . . . . . . . . . . . . . . . . . . . .
+42
+4
+Projection groupoids
+43
+4.1
+Projection groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+43
+4.2
+Chained projection groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+47
+4.3
+The regular ∗-semigroup associated to a chained projection groupoid . . . . . . .
+51
+4.4
+Products of projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+55
+5
+Regular ∗-semigroups
+57
+5.1
+The chained projection groupoid associated to a regular ∗-semigroup . . . . . . .
+58
+5.2
+Examples
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+62
+6
+The category isomorphism
+66
+II
+Applications
+71
+7
+Idempotent-generated regular ∗-semigroups
+71
+7.1
+The chain semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+72
+7.2
+Presentation by generators and relations . . . . . . . . . . . . . . . . . . . . . . .
+76
+7.3
+Chain semigroups as free objects
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+79
+8
+Fundamental regular ∗-semigroups
+83
+8.1
+The maximum idempotent-separating congruence . . . . . . . . . . . . . . . . . .
+83
+8.2
+Maximum fundamental regular ∗-semigroups . . . . . . . . . . . . . . . . . . . . .
+85
+8.3
+Idempotent-generated fundamental regular ∗-semigroups . . . . . . . . . . . . . .
+91
+9
+Inverse semigroups and inductive groupoids
+92
+9.1
+The chained projection groupoid associated to an inverse semigroup
+. . . . . . .
+93
+9.2
+The Ehresmann–Schein–Nambooripad Theorem . . . . . . . . . . . . . . . . . . .
+94
+9.3
+Fundamental inverse semigroups
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+97
+2
+
+1
+Introduction
+‘All of mathematics is the study of symmetry’ — H.S.M. Coxeter.
+The broadest possible setting in which to place the current work is in the study of mathematical
+symmetry. This should practically go without saying, given the above quote, attributed to Cox-
+eter by Roberts in [125]. Nevertheless, the algebraic and categorical structures we are concerned
+with had their origins in the concerted efforts of mathematicians and scientists to pin down the
+exact meaning of ‘symmetry’, and to devise the best ways to encode and study it.
+We could not hope to adequately recount the fascinating history here, but fortunately we
+can instead refer readers to the introduction of Lawson’s monograph [87], and also to the work
+of Hollings [66]. We pick up the story in the late nineteenth century, long after the discovery
+of group theory, when mathematicians were facing a plethora of new non-Euclidean geometries.
+Some of these had trivial automorphism groups, yet were intuitively highly symmetric. Two main
+approaches to capture this non-global or partial symmetry eventually emerged in the middle of
+the twentieth century: the inductive groupoids of Charles Ehresmann [45, 46], and the inverse
+semigroups of Viktor Wagner [131, 132] and Gordon Preston [121, 122]. Far from being rival
+approaches, the two theories were eventually united in one of the landmark results of inverse
+semigroup theory:
+Ehresmann–Schein–Nambooripad Theorem. The categories of inverse semigroups (with
+semigroup homomorphisms) and inductive groupoids (with inductive functors) are isomorphic.
+This theorem contains too much information to fully unpack just now, but more will be said in
+Chapter 9. The ESN Theorem was first formulated by Lawson in [87, Theorem 4.1.8], who named
+it after the three mathematicians who had contributed most to its development. We have already
+encountered Ehresmann. The second named author, Boris Schein, was largely responsible for
+establishing the explicit connection between inverse semigroups and inductive groupoids [128].
+The third, K.S. Subramonian Nambooripad [110], was credited for his categorical framework
+(albeit in a different context), wherein the theorem is couched in terms of a relationship between
+entire categories (including their morphisms) of semigroups and groupoids, rather than simply
+showing how the algebraic and categorical structures are inter-definable.
+The ESN Theorem has had many far-reaching consequences, both within the fields of inverse
+semigroups and inductive groupoids (see [87]), and also beyond. Before we comment on the
+latter, it is worth addressing the former, by confronting an elephant in the room:
+• Inverse semigroups are relatively simple mathematical objects. In what sense does it ‘help’
+to view them through the lens of groupoids, and category theory?
+The precise answer to this question would again take us too far afield, but some brief comments
+should suffice for the time being. First, an inverse semigroup S can in principle be completely
+understood in terms of its multiplication table. Each pair of elements a, b ∈ S gives rise to the
+product ab ∈ S. This binary operation must be associative, i.e. satisfy the law (ab)c = a(bc).
+Further, each element a ∈ S must have a unique (semigroup-theoretic) inverse in S, i.e. an
+element a′ ∈ S satisfying a = aa′a and a′ = a′aa′. Uniqueness means that we can think of
+a �→ a′ as an additional (unary) operation of S, and we generally write a′ = a−1.
+Among
+other things, the ESN Theorem says that one can associate a groupoid G = G(S) to such an
+inverse semigroup S. We will postpone the full definition of G for now, but two key points bear
+emphasising:
+• the morphisms of G are precisely the elements of S, and
+• the (partial) composition in G is a restriction of the (total) product in S.
+3
+
+Roughly speaking, the groupoid G ‘remembers’ only the ‘easy’ products of S, and the ESN
+Theorem tells us that (combined with some order-theoretic data) these easy products are enough
+to reconstruct the entire multiplication table of S. (The exact meaning of ‘easy’ will be explored
+a little below, and in more detail in Sections 2.3 and 2.4.) In a sense, the latter point encapsulates
+the deepest part of the theorem: Inductive groupoids are precisely the (abstract) categories G
+whose composition can be extended in a natural way to construct an inverse semigroup S = S(G).
+Moreover, the constructions S �→ G(S) and G �→ S(G) are mutually inverse functors between
+the categories of inverse semigroups and inductive groupoids. This category isomorphism is not
+merely a convenient way to say that inverse semigroups and inductive groupoids are ‘one and
+the same’, however. The fact that the isomorphism sends morphisms to morphisms means that
+a map φ : S → S′ between inverse semigroups is a semigroup homomorphism if and only if it is
+a (so-called) inductive functor G(φ) = φ : G(S) → G(S′). Less effort is required to show that φ is
+a functor, as opposed to a full semigroup homomorphism, as the former involves checking that
+(a ◦ b)φ = (aφ) ◦ (bφ)
+for composable pairs a, b ∈ G(S) ≡ S,
+and we emphasise that the composable pairs in G are precisely the easy products in S.
+The ESN Theorem has had a major impact outside of inverse semigroup theory, with one
+particularly fruitful direction being the application of inverse semigroups to C∗-algebras. For
+example, the articles [12,14,47,80,83,89,94,95,105,117] display a mixture of semigroup-theoretic
+and groupoid-based approaches. For more extensive discussions, and many more references, see
+Lawson’s survey [90] and Paterson’s monograph [116].
+Another way to measure the influence of the ESN Theorem is to consider its extensions and
+generalisations. When faced with an important class of semigroups, one naturally looks for an
+‘ESN-type link’ with an equally-natural class of categories, and vice versa. Thus, we have ESN-
+type theorems for regular semigroups, restriction semigroups, Ehresmann semigroups, concordant
+semigroups, and others; see for example [1, 18, 49, 50, 53–55, 65, 66, 86, 88, 91, 110, 133–135]. As
+with other ‘dualities’ in mathematics [8, 15, 119, 120, 123, 124, 129], such correspondences allow
+problems in one field to be translated into another, where they can hopefully be solved, and the
+solution then reinterpreted in the original context.
+Arguably the most significant of the papers just cited was Nambooripad’s 1979 memoir [110].
+As well as pioneering the categorical approach (which led to the ‘N’ in ‘ESN’), this paper majorly
+extended the ESN Theorem to the class of regular semigroups. These are semigroups in which
+every element has at least one inverse. Dropping the uniqueness restriction on inverses leads
+to a far more general class of semigroups, which contains many additional natural examples,
+including semigroups of mappings, linear transformations, and more. The increase in generality
+of the semigroups led inevitably to an increase in the complexity of the categorical structures
+modelling them. Although Nambooripad still represented a regular semigroup S by an ordered
+groupoid G = G(S), the groupoid G was not enough to completely recover the semigroup; an
+additional layer of structure was required.
+The need for this extra data is due to the fact
+that the object set of G is not a semilattice (as for inverse semigroups), but instead a (regular)
+biordered set: a partial algebra with a pair of intertwined pre-orders satisfying a fairly complex
+set of axioms. The increase in generality also led to the sacrifice of the category isomorphism.
+Nambooripad’s main result, [110, Theorem 4.14], is that the category of regular semigroups is
+equivalent to a certain category of groupoids (which Nambooripad also called inductive).
+One way to understand this sacrifice is via the loss of symmetry when moving from inverse
+to regular semigroups. As we discussed above, inverse semigroups were devised to model partial
+symmetries that were unrecognisable by groups.
+This is formalised in the Wagner–Preston
+Theorem, which states that any inverse semigroup is (isomorphic to) a semigroup of partial
+symmetries of some mathematical structure; see [87, Theorem 1.5.1]. But an inverse semigroup
+is itself an extremely symmetrical mathematical structure in its own right, so much so that the
+canonical proof of the Wagner–Preston Theorem involves showing that an inverse semigroup
+4
+
+is (isomorphic to) a semigroup of partial symmetries of itself !
+This ‘internal symmetry’ is
+manifested in the properties of the inversion operation a �→ a−1, speficially the fact that this is
+an involution, i.e. satisfies the familiar laws
+(a−1)−1 = a
+and
+(ab)−1 = b−1a−1,
+(1.1)
+beyond the defining property that a and a−1 are mutual inverses:
+a = aa−1a
+and
+a−1 = a−1aa−1.
+(1.2)
+Inverse semigroups actually satisfy an additional law:
+aa−1 · bb−1 = bb−1 · aa−1,
+(1.3)
+and this is in fact equivalent to the commutativity of idempotents. This in turn means that the
+set E = E(S) of idempotents of an inverse semigroup S forms a semilattice, with the (order-
+theoretic) meet of two idempotents being their (semigroup) product: e∧f = ef. This then feeds
+into the definition of the groupoid G = G(S). The object set of G is E, and each element a ∈ S
+is thought of as a morphism from a ‘domain idempotent’ d(a) = aa−1 to a ‘range idempotent’
+r(a) = a−1a. Beyond the obvious law
+d(a) · a = a = a · r(a),
+(1.4)
+which follows immediately from (1.2), the key point making everything work is that
+r(a) = d(b)
+⇒
+d(ab) = d(a) and r(ab) = r(b).
+(1.5)
+This allows us to define the composition
+a ◦ b = ab
+when r(a) = d(b).
+(1.6)
+These are the ‘easy’ products alluded to above. Going in the reverse direction, the semilattice
+structure of the object set E of an inductive groupoid G is crucial in extending the (partial)
+composition to a (total) product. Given two morphisms a, b ∈ G, the range e = r(a) and domain
+f = d(b) have a meet (greatest lower bound) in E: g = e ∧ f. The ordered structure of G means
+that we have (right and left) restrictions a′ = a⇂g and b′ = g⇃b, and since r(a′) = g = d(b′), these
+are composable in G, so that we can define
+a • b = a′ ◦ b′ = a⇂g ◦ g⇃b,
+and in this way reconstruct the entire structure of S = S(G). The operation • is often called the
+pseudo-product in the literature.
+The major difficulty in extending the above ideas to arbitrary regular semigroups is in the
+loss of uniqueness of inverses, and hence in the possible expansion of the available domain and
+range idempotents. Indeed, if an element a of a regular semigroup S possesses distinct inverses
+a′, a′′ ∈ S, then one is faced with a dilemma: whether to think of a as a morphism aa′ → a′a
+or aa′′ → a′′a, and in general it is quite possible to have aa′ ̸= aa′′ and/or a′a ̸= a′′a. Thus,
+Nambooripad was led to consider the pairs (a, a′) and (a, a′′) as separate morphisms, essentially
+splitting each semigroup element into several morphisms, one for each inverse, and then consider-
+ing equivalence-classes of such pairs. This also necessitated a far more elaborate pseudo-product.
+Given semigroup elements a, b ∈ S, one would like to identify suitable domain and range idem-
+potents, e = r(a) and f = d(b), and then to find composable restrictions a⇂g and g⇃b (with g
+‘below’ e and f) in order to define a • b = a⇂g ◦ g⇃b, as above. This is not generally possible,
+however, and one inevitably has to work with the biordered sets alluded to above. Specifically,
+Nambooripad located g in the so-called sandwich set S(e, f). The restrictions a⇂g and g⇃b still
+5
+
+may or may not exist, but a⇂eg and gf⇃b do. These latter restrictions still may or may not be
+composable, but it is possible to find a distinguished element x for which the composition
+a⇂eg ◦ x ◦ gf⇃b
+does exist, and can be taken as the definition of the pseudo-product a • b. This element x has
+the form
+x = ε(eg, g) ◦ ε(g, gf) = ε(eg, g, gf),
+where ε is a special functor into G from a certain category of ‘E-chains’. In the end, Namboori-
+pad’s ‘groupoids’ are in fact pairs (G, ε), where G is a special ordered groupoid with a biordered
+object set, and where ε is a special functor from the E-chain groupoid into G. The ingredients G
+and ε are linked via a certain coherence condition, and a property of so-called singular squares
+in the biordered set. It would be impossible to adequately cover all the details and subtleties of
+Nambooripad’s work, so instead we refer the reader to the survey [104]. In any case, the step up
+in complexity from the inverse case should be apparent from our brief treatment.
+Let us now take a small step back. Recall that inverse semigroups are the semigroups with
+a unary operation a �→ a−1 satisfying the laws (1.1), (1.2) and (1.3). Associated to such an
+inverse semigroup S, one can define a groupoid G = G(S) with morphism set S, and with
+composition (1.6). As it happens, law (1.3) is not a necessary ingredient in the construction of
+the groupoid G. In other words, if S is a semigroup satisfying laws (1.1) and (1.2), then the above
+procedure works without modification to produce a groupoid G = G(S). If law (1.3) does not
+hold, then the groupoid may or may not be inductive, but it still has many remarkable properties
+shared by inductive groupoids. This observation is, in essence, the starting point for the current
+work.
+Semigroups satisfying the laws (1.1) and (1.2) are precisely the regular ∗-semigroups of Nor-
+dahl and Scheiblich [114]. (These semigroups are not to be confused with the ∗-regular semigroups
+of Drazin [26], and are known by some other names in the literature; for example, they are called
+special ∗-semigroups in [113].) As the terminology suggests, the involution on such a semigroup
+is generally denoted a �→ a∗ (reserving a−1 for inverse semigroups), and this must satisfy the
+laws
+(a∗)∗ = a = aa∗a
+and
+(ab)∗ = b∗a∗.
+The class of regular ∗-semigroups has received a great deal of attention in recent years, partly
+because of the prototypical examples of the so-called diagram semigroups. These semigroups
+include families such as the Brauer, Temperley-Lieb, Kauffman and partition monoids, and have
+their origins and applications in a wide range of mathematical and scientific disciplines, from
+representation theory, low-dimensional topology, statistical mechanics, and many more; see for
+example [2–4, 6, 7, 9, 10, 20, 21, 39–41, 43, 44, 51, 62, 74–79, 81, 82, 85, 92, 93, 97–103, 130, 136, 137].
+These monoids have provided a strong bridge between semigroup theory and these other branches
+of mathematics; for a fuller discussion of this fruitful dialogue, and for many more references,
+see the introduction to [39].
+As we will see, regular ∗-semigroups occupy something of a ‘sweet spot’ between the classes of
+inverse and regular semigroups. On the one hand, their involution provides the ‘internal symme-
+try’ possessed by inverse semigroups, allowing for the natural groupoid representations discussed
+above. On the other hand, the non-commutativity of idempotents (i.e. the dropping of law (1.3))
+leads to a much richer idempotent structure, which in turn means that the groupoid representa-
+tion is not faithful, and necessitates a more intricate approach when defining the pseudo-product.
+In a way, this ‘balance’ can be summed up in an observation about the proliferation of inverses.
+In an inverse semigroup, the inverse ‘picks itself’, in the sense that each element has exactly one.
+In a regular semigroup there is some degree of ‘chaos’; while inverses are guaranteed to exist,
+there could be an over-abundance (there even exist regular (∗-)semigroups in which every ele-
+ment is an inverse of every other element). In regular ∗-semigroups, inverses are not necessarily
+6
+
+unique, but the involution ‘picks one for you’. The involution also leads to a canonical way to
+‘pick’ domain and range idempotents; for a ∈ S, one can take
+d(a) = aa∗
+and
+r(a) = a∗a.
+(1.7)
+Such idempotents have the additional property that they are invariant under the involution, as
+for example (aa∗)∗ = (a∗)∗a∗ = aa∗. Such an idempotent is called a projection, and the set of
+all such projections is denoted
+P = P(S) = {p ∈ S : p2 = p = p∗}.
+This is of course contained in the set
+E = E(S) = {e ∈ S : e2 = e}
+of all idempotents, but we do not have equality in general; in fact, it is well known (and easy to
+see) that the regular ∗-semigroup S is inverse if and only if E = P. Although P = P(S) is not a
+subsemigroup of the regular ∗-semigroup S in general, it can be given an algebraic structure via
+the ‘conjugation action’ of projections on each other. Each element p ∈ P determines a unary
+operation
+θp : P → P
+given by
+qθp = pqp
+for q ∈ P.
+These so-called projection algebras have been used by Imaoka (under a different name) and
+Jones (in a different, but equivalent, form) in [72] and [73], and are the appropriate ‘∗-analogues’
+of semilattices in the theory of regular ∗-semigroups. (Semilattices are indeed a special case,
+wherein we define eθf = e ∧ f.) In particular, the groupoid G = G(S) associated to a regular
+∗-semigroup S has the projection algebra P = P(S) for its object set. Moreover, (1.4) and (1.5)
+both hold (with respect to the domains and ranges given in (1.7)), allowing us to define the
+composition in G exactly as in (1.6). Conversely, in attempting to define a pseudo-product •
+from such a ‘projection groupoid’ G, one first identifies the range p = r(a) and domain q = d(b),
+calculates two further projections p′ = qθp and q′ = pθq (which collectively play the role of a
+meet p ∧ q, which may or may not exist), and then forms the restrictions a⇂p′ and q′⇃b. These
+are still not composable in general, as we do not always have p′ = q′, but there always exists a
+special morphism e : p′ → q′, allowing us to define
+a • b = a⇂p′ ◦ e ◦ q′⇃b.
+This morphism e is again in the image of a special functor ε : C → G, where here C = C (P) is a
+certain ‘chain groupoid’ associated to the projection algebra P. This all results in what we call
+a chained projection groupoid, which is a pair (G, ε), consisting of:
+• an ordered groupoid G, whose object set P is an (abstract) projection algebra, with strong
+links between the categorical and algebraic structures of G and P, and
+• a functor ε : C = C (P) → G obeying a natural coherence condition.
+Roughly speaking, our main result (Theorem 6.7) states that this construction is invertible,
+and furnishes an isomorphism between the categories of regular ∗-semigroups and (abstract)
+chained projection groupoids. Specialising to the inverse case, we obtain the ESN Theorem as
+a corollary. We emphasise, however, that Theorem 6.7 is not a specialisation of Nambooripad’s
+result on regular semigroups [110].
+In fact, such a specialisation would invariably lead to a
+category equivalence (to a very different category of groupoids) rather than an isomorphism, as
+inverses are generally not unique in regular ∗-semigroups.
+Any further discussion would necessitate additional technicalities that are undesirable at this
+stage. So instead, let us now give a brief summary of the structure of the article. After this
+introductory chapter, we begin in Chapter 2 with preliminaries on (regular ∗-)semigroups and
+ordered categories, as well as a fairly detailed discussion of the special case of partition monoids.
+The paper then splits into two main parts:
+7
+
+• Part I is devoted to the proof of our main result, Theorem 6.7, which establishes the isomor-
+phism between the categories of regular ∗-semigroups and chained projection groupoids.
+• Part II then applies the machinery developed in the first part to (free) idempotent-generated
+regular ∗-semigroups, fundamental regular ∗-semigroups and inverse semigroups.
+Part I contains Chapters 3–6, which build towards our main structural result, stated in Theo-
+rem 6.7.
+• Chapter 3 begins with the most basic building block, the so-called projection algebras. In
+particular, we show how to build a number of categorical structures on top of an abstract
+projection algebra P, culminating in the construction of the chain groupoid C = C (P).
+• In Chapter 4 we turn to the class of projection groupoids. These are ordered groupoids G,
+whose object set P is a projection algebra, with strong links between the structures of G (as a
+groupoid) and P (as an algebra). The key idea is that of a chained projection groupoid, which
+is a pair (G, ε) consisting of a projection groupoid G, and a special functor ε : C → G called
+an evaluation map, where here C = C (P) is the chain groupoid of the projection algebra P.
+The main result of the chapter is Theorem 4.36, which shows how to construct a regular
+∗-semigroup S = S(G, ε) from a chained projection groupoid (G, ε).
+• Chapter 5 goes in the opposite direction, and shows how a regular ∗-semigroup S gives rise to
+a chained projection groupoid (G, ε) = G(S); see Theorem 5.19. In a sense, this breaks down
+the structure of S into simpler parts: the groupoid G remembers only the ‘easy products’
+in S; the projection algebra P remembers the projections of S and their conjugation action;
+and the evaluation map ε tells us how these parts fit together. This decomposition can be
+thought of as a structure theorem for arbitrary regular ∗-semigroups (see Remark 6.8); as far
+as we are aware it is the first of its kind.
+• We then bring these ideas together in Chapter 6 in order to complete the proof of Theorem 6.7.
+This involves showing that the S and G constructions are in fact mutually inverse functors
+between the categories of regular ∗-semigroups and chained projection groupoids.
+Part II contains Chapters 7–9, which comprise a number of applications of the theory developed
+in Part I.
+• Chapter 7 concerns idempotent-generated regular ∗-semigroups. The main construction here is
+the so-called chain semigroup associated to an arbitrary (abstract) projection algebra. These
+semigroups enjoy several categorical ‘free-ness’ properties, as described in Theorems 7.11, 7.20
+and 7.23. Theorem 7.14 gives a presentation by generators and relations.
+• We then consider fundamental regular ∗-semigroups in Chapter 8. These have been previously
+studied by a number of authors [70,72,113,138], but we show how our general machinery leads
+to a new way to build these semigroups; see Theorems 8.20 and 8.21. In Theorem 8.24 we
+show that (up to isomorphism) there is a unique idempotent-generated fundamental regular
+∗-semigroup with a given (abstract) projection algebra, and show how to construct it.
+• Finally, in Chapter 9 we show how the entire theory simplifies in the case of inverse semi-
+groups. As an important application, we give a new proof of the Ehresmann–Schein–Nambooripad
+Theorem, stated above.
+Throughout the paper, we consider a number of examples, which serve both to illustrate the
+general theory, and also to highlight some of the subtleties that arise. For the same reasons, at
+times we will compare our constructions and results with the regular and inverse cases. We also
+pose a number of open problems, which we believe are worthy of further study.
+8
+
+2
+Preliminaries
+In this chapter we gather the preliminary definitions and basic results we require in the rest of
+the paper, and establish notation. We begin in Section 2.1 with the main ideas from semigroup
+theory, and then in Section 2.2 we discuss (ordered) categories and groupoids.
+Section 2.3
+contains some basic background on regular ∗-semigroups, as well as a foreshadowing of various
+ideas that will occur throughout the paper.
+This material is included to motivate the more
+abstract results of later chapters. Finally, Section 2.4 concerns a particularly important concrete
+class of regular ∗-semigroups, namely the partition monoids.
+This is again included to give
+more motivation for our later ideas and results, and to show how these can be understood
+diagrammatically for these monoids. We also comment on a number of known results, and show
+how they can be interpreted through the groupoid lens.
+For further general background, we refer the reader to texts such as [16,68] for semigroups, [87]
+for inverse semigroups, [5,96] for categories, and [13] for universal algebra, though the latter is
+only used tangentially.
+2.1
+Semigroups
+A semigroup is a set with an associative binary operation, which will typically be denoted by
+juxtaposition. A monoid is a semigroup with an identity element. As usual we write S1 for the
+monoid completion of the semigroup S. So S1 = S if S happens to be a monoid; otherwise,
+S1 = S ∪ {1}, where 1 is a symbol not belonging to S, acting as an adjoined identity element.
+Green’s relations are five equivalence relations, defined on an arbitrary semigroup S as fol-
+lows [59]. First, for a, b ∈ S we have
+a R b ⇔ aS1 = bS1,
+a L b ⇔ S1a = S1b,
+a J b ⇔ S1aS1 = S1bS1.
+Note that a R b precisely when either a = b or else a = bx and b = ay for some x, y ∈ S; similar
+comments apply to L and J . The remaining two of Green’s relations are defined by
+H = R ∩ L
+and
+D = R ∨ L ,
+where the latter is the join in the lattice of equivalences on S, i.e. the least equivalence contain-
+ing R ∪ L . It is well known that D = R ◦ L = L ◦ R, and that D = J if S is finite. We
+denote the R-class of a ∈ S by
+Ra = {b ∈ S : a R b},
+and similarly for L -classes La, and so on.
+An element a of a semigroup S is regular if a = axa for some x ∈ S. The element y = xax
+then has the property that a = aya and y = yay; such an element y is called an inverse of x.
+We say S is regular if every element of S is regular. The set of idempotents of S is denoted
+E(S) = {e ∈ S : e = e2}.
+Idempotents are of course regular. An inverse semigroup is a semigroup in which every element
+has a unique inverse. It is well known that S is inverse if and only if it is regular and its idempo-
+tents commute; in this case, E(S) is a semilattice, i.e. a semigroup of commuting idempotents.
+In this paper we are concerned with a class of semigroups contained strictly between regular
+and inverse semigroups, the so-called regular ∗-semigroups of Nordahl and Scheiblich [114]. We
+postpone their definition until Section 2.3.
+A congruence on a semigroup S is an equivalence relation σ that is compatible with the
+product of S, meaning that
+a σ b
+⇒
+ax σ bx and xa σ xb
+for all a, b, x ∈ S,
+9
+
+or equivalently that
+a σ b and x σ y
+⇒
+ax σ by
+for all a, b, x, y ∈ S.
+Given a congruence σ, the quotient semigroup S/σ consists of all σ-classes under the induced
+product, [a][b] = [ab]. The kernel of a semigroup homomorphism φ : S → T is the congruence
+ker(φ) = {(a, b) ∈ S × S : aφ = bφ}.
+The fundamental homomorphism theorem for semigroups says that the quotient S/ ker(φ) is
+isomorphic to im(φ), the image of S under φ.
+2.2
+Categories
+Unless stated otherwise, the categories we are concerned with are assumed to be small. We
+typically identify a (small) category C with its set of morphisms. We identify the objects of C with
+the identities, the set of which is denoted vC. We denote the domain and codomain (a.k.a. range)
+of a ∈ C by d(a) and r(a), respectively. We compose morphisms left to right, so a◦b is defined if
+and only if r(a) = d(b), in which case d(a◦b) = d(a) and r(a◦b) = r(b). (Sometimes functors will
+be written to the left of their arguments, and so composed right to left; it should always be clear
+which convention is being used.) For p, q ∈ vC, we write C(p, q) = {a ∈ C : d(a) = p, r(a) = q}
+for the set of all morphisms p → q.
+All the (small) categories we study will have an involution and an order, as made precise in
+the next two standard definitions.
+Definition 2.1. A ∗-category is a (small) category C with an involution, i.e. a map C → C : a �→ a∗
+satisfying the following, for all a, b ∈ C:
+(I1) d(a∗) = r(a) and r(a∗) = d(a).
+(I2) (a∗)∗ = a.
+(I3) If r(a) = d(b), then (a ◦ b)∗ = b∗ ◦ a∗.
+A groupoid is a ∗-category for which we additionally have
+(I4) a ◦ a∗ = d(a) (and hence also a∗ ◦ a = r(a)) for all a ∈ C.
+In a groupoid, we typically write a∗ = a−1 for a ∈ C.
+It is easy to show that p∗ = p for all p ∈ vC, when C is a ∗-category. It is also clear that (I1)
+is a consequence of (I4), so a groupoid is a category with a map a �→ a∗ satisfying (I2)–(I4).
+Definition 2.2. An ordered ∗-category is a ∗-category C equipped with a partial order ≤ satis-
+fying the following, for all a, b, c, d ∈ C and p ∈ vC:
+(O1) If a ≤ b, then d(a) ≤ d(b) and r(a) ≤ r(b).
+(O2) If a ≤ b, then a∗ ≤ b∗.
+(O3) If a ≤ b and c ≤ d, and if r(a) = d(c) and r(b) = d(d), then a ◦ c ≤ b ◦ d.
+(O4) For all p ≤ d(a), there exists a unique u ≤ a with d(u) = p.
+It is easy to see that (O1)–(O4) imply the following, which is a dual of (O4):
+10
+
+(O4)∗ For all q ≤ r(a), there exists a unique v ≤ a with r(v) = q.
+(Here we have q ≤ d(a∗), and if w is the unique element with w ≤ a∗ with d(w) = q, then we
+take v = w∗.)
+The elements u and v in (O4) and (O4)∗ are denoted u = p⇃a and v = a⇂q, respectively, and
+called the left restriction of a to p and the right restriction of a to q. Some authors call a⇂q a
+restriction, and p⇃a a co-restriction; we prefer the left/right terminology, however, as we feel that
+it does not ‘prioritise’ one over the other.
+An ordered groupoid is a groupoid with a partial order satisfying (O1)–(O4). In fact, when C
+is a groupoid, (O2), (O3) and (I4) together imply (O1).
+It is easy to see that the object set vC is an order ideal in any ordered ∗-category C, meaning
+that the following holds:
+• For all a ∈ C and p ∈ vC, a ≤ p ⇒ a ∈ vC.
+Indeed, if a ≤ p, and if we write q = d(a) ∈ vC, then by (O1) we have q ≤ d(p) = p. But then
+a, q ≤ p and d(a) = q = d(q), so by uniqueness in (O4) we have a = q ∈ vC. It also follows from
+this that q⇃p = q for any q ≤ p. We typically use facts such as these without explicit reference.
+In what follows, it is typically more convenient to construct an ordered ∗-category C by:
+• defining an order ≤ on the object set vC,
+• defining left restrictions p⇃a, for a ∈ C and p ≤ d(a),
+• specifying that a ≤ b (for morphisms a, b ∈ C) if a is a restriction of b.
+The next lemma axiomatises the conditions required to ensure that we do indeed obtain an
+ordered ∗-category in this way.
+Lemma 2.3. Suppose C is a ∗-category for which the following two conditions hold:
+(i) There is a partial order ≤ on the object set vC.
+(ii) For all a ∈ C, and for all p ≤ d(a), there exists a morphism p⇃a ∈ C, such that the following
+hold, for all a, b ∈ C and p, q ∈ vC:
+(O1)′ If p ≤ d(a), then d(p⇃a) = p and r(p⇃a) ≤ r(a).
+(O2)′ If p ≤ d(a), and if q = r(p⇃a), then (p⇃a)∗ = q⇃a∗.
+(O3)′ d(a)⇃a = a.
+(O4)′ For all p ≤ q ≤ d(a), we have p⇃q⇃a = p⇃a.
+(O5)′ If p ≤ d(a) and r(a) = d(b), and if q = r(p⇃a), then p⇃(a ◦ b) = p⇃a ◦ q⇃b.
+Then C is an ordered ∗-category with order given by
+a ≤ b
+⇔
+a = p⇃b
+for some p ≤ d(b).
+(2.4)
+Moreover, any ordered ∗-category has the above form.
+Proof. Beginning with the final assertion, suppose C is an ordered ∗-category. Then vC is a
+sub-poset of C, and hence (i) holds. For (ii), we take p⇃a to be the morphism u ≤ a from (O4),
+and (O1)′–(O5)′ are all easily checked. For example, to verify (O2)′, suppose a ∈ C and p ≤ d(a),
+and let q = r(p⇃a).
+Then since p⇃a ≤ a, it follows from (O2) that (p⇃a)∗ ≤ a∗, and we
+11
+
+have d((p⇃a)∗) = r(p⇃a) = q. But q⇃a∗ is the unique element below a∗ with domain q, so in
+fact (p⇃a)∗ = q⇃a∗.
+Conversely, suppose conditions (i) and (ii) both hold. We first check that the relation ≤
+in (2.4) is a partial order. Reflexivity follows immediately from (O3)′, and transitivity from (O4)′.
+For anti-symmetry, suppose a ≤ b and b ≤ a, so that a = p⇃b and b = q⇃a for some p ≤ d(b) and
+q ≤ d(a). Then
+a = p⇃b = p⇃q⇃a = p⇃a
+⇒
+d(a) = d(p⇃a) = p ≤ d(b)
+and similarly
+d(b) = q ≤ d(a).
+It follows that p = q = d(a) = d(b), and so a = p⇃b = d(b)⇃b = b. Now that we know ≤ is a
+partial order, we verify conditions (O1)–(O4).
+(O1) and (O2). Suppose a ≤ b, so that a = p⇃b for some p ≤ d(b). Then (O1)′ gives
+d(a) = d(p⇃b) = p ≤ d(b)
+and
+r(a) = r(p⇃b) ≤ r(b).
+We also have a∗ = (p⇃b)∗ = q⇃b∗ by (O2)′, where q = r(p⇃b), so that a∗ ≤ b∗.
+(O3). Suppose a ≤ b and c ≤ d are such that r(a) = d(c) and r(b) = d(d). So a = p⇃b and
+c = q⇃d for some p ≤ d(b) and q ≤ d(d). Since
+q = d(q⇃d) = d(c) = r(a) = r(p⇃b),
+it follows from (O5)′ that a ◦ c = p⇃b ◦ q⇃d = p⇃(b ◦ d), so that a ◦ c ≤ b ◦ d.
+(O4). Given p ≤ d(a), we certainly have u ≤ a and d(u) = p, where u = p⇃a. For uniqueness,
+suppose also that x ≤ a for some x ∈ C with d(x) = p. Since x ≤ a, we have x = q⇃a for some
+q ≤ d(a). But then q = d(q⇃a) = d(x) = p, so in fact x = q⇃a = p⇃a = u.
+Remark 2.5. The previous result referred only to left restrictions. Right restrictions can be
+defined from the left, using the involution:
+a⇂q = (q⇃a∗)∗
+for a ∈ C and q ≤ r(a).
+As explained in Definition 2.2, a⇂q is the unique element below a with codomain q. In particular,
+the following holds in any ordered ∗-category:
+(O6)′ If a ∈ C and p ≤ d(a), and if q = r(p⇃a), then p⇃a = a⇂q.
+Each of (O1)′–(O6)′ of course have duals. For example, the dual of (O5)′ says:
+• If q ≤ r(b) and r(a) = d(b), and if p = d(b⇂q), then (a ◦ b)⇂q = a⇂p ◦ b⇂q.
+It is also worth noting that for a, b ∈ C we have
+a ≤ b ⇔ a = p⇃b
+for some p ≤ d(b)
+⇔ a = b⇂q
+for some q ≤ r(b)
+⇔ a = p⇃b = b⇂q
+for some p ≤ d(b) and q ≤ r(b).
+The p, q ∈ P here are of course p = d(a) and q = r(a).
+12
+
+Definition 2.6. A v-congruence on a category C is an equivalence relation ≈ on C satisfying
+the following, for all a, b, u, v ∈ C:
+(C1) a ≈ b ⇒ [d(a) = d(b) and r(a) = r(b)],
+(C2) a ≈ b ⇒ u ◦ a ≈ u ◦ b, whenever the stated compositions are defined,
+(C3) a ≈ b ⇒ a ◦ v ≈ b ◦ v, whenever the stated compositions are defined.
+If C is a ∗-category, we say that ≈ is a ∗-congruence if it satisfies (C1)–(C3) and the following,
+for all a, b ∈ C:
+(C4) a ≈ b ⇒ a∗ ≈ b∗.
+If C is an ordered ∗-category, we say that ≈ is an ordered ∗-congruence if it satisfies (C1)–(C4)
+and the following, for all a, b ∈ C and p ∈ vC:
+(C5) [a ≈ b and p ≤ d(a)] ⇒
+p⇃a ≈ p⇃b.
+Given a v-congruence ≈ on a category C, the quotient category C/≈ consists of all ≈-classes,
+under the induced composition.
+We typically write [a] for the ≈-class of a ∈ C.
+It follows
+immediately from (C1) that p ≈ q ⇒ p = q for objects p, q ∈ vC, so we can identify the object
+sets of C and C/≈, viz. p ≡ [p]. In this way, for a ∈ C we have d[a] = d(a) and r[a] = r(a) for all
+a ∈ C, and
+[a] ◦ [b] = [a ◦ b]
+whenever r[a] = d[b].
+Lemma 2.7.
+(i) If ≈ is a ∗-congruence on a ∗-category C, then C/≈ is a ∗-category, with
+involution given by
+[a]∗ = [a∗]
+for all a ∈ C.
+(2.8)
+If also a ◦ a∗ ≈ d(a) for all a ∈ C, then C/≈ is a groupoid.
+(ii) If ≈ is an ordered ∗-congruence on an ordered ∗-category C, then C/≈ is an ordered ∗-
+category, with involution (2.8), and order given by
+α ≤ β
+⇔
+a ≤ b
+for some a ∈ α and b ∈ β.
+(2.9)
+Proof. Part (i) is routine, and (ii) only a little more difficult, so we just give a sketch for the
+latter.
+For this, one begins by showing that α ≤ β is equivalent to the ostensibly stronger
+condition:
+• For all b ∈ β, there exists a ∈ α such that a ≤ b.
+It is then easy to check that ≤ is indeed a partial order on C/≈. Conditions (O1)–(O4) for C/≈
+follow quickly from the corresponding conditions for C.
+We will shortly return to congruences in Lemma 2.17 below, where we give simpler criteria
+for checking that a v-congruence satisfies (C4) or (C5). The statement of the lemma will require
+certain maps defined on any ordered ∗-category. These maps will also be used extensively in the
+rest of the paper. To define them, fix some such ordered ∗-category C with object set P = vC.
+For p ∈ P, we write
+p↓ = {q ∈ P : q ≤ p}
+for the down-set of p in the poset (P, ≤). Consider a morphism a ∈ C. A restriction p⇃a is defined
+precisely when p ≤ d(a), and we write pϑa = r(p⇃a) for the codomain of p⇃a; by (O1)′, we have
+pϑa ≤ r(a). In other words, we have a map
+ϑa : d(a)↓ → r(a)↓
+given by
+pϑa = r(p⇃a).
+(2.10)
+13
+
+Note that for q ≤ r(a), we have
+d(a⇂q) = d((q⇃a∗)∗) = r(q⇃a∗) = qϑa∗.
+(2.11)
+Lemma 2.12. If C is an ordered ∗-category, then for any a ∈ C,
+ϑa : d(a)↓ → r(a)↓
+and
+ϑa∗ : r(a)↓ → d(a)↓
+are mutually inverse bijections.
+Proof. By symmetry, it is enough to show that ϑaϑa∗ = idd(a)↓. To do so, let p ≤ d(a), and
+write q = pϑa = r(p⇃a). By (O6)′ we have p⇃a = a⇂q. Combining this with (O1)′ and (2.11), we
+deduce that
+p = d(p⇃a) = d(a⇂q) = qϑa∗ = pϑaϑa∗,
+as required.
+Note that property (O5)′ and its dual (cf. Remark 2.5) can be rephrased in terms of the ϑ
+maps. Specifically, if a, b ∈ C (an ordered ∗-category) are such that r(a) = d(b), then (keep-
+ing (2.11) and Lemma 2.12 in mind) we have
+p⇃(a◦b) = p⇃a◦pϑa⇃b
+and
+(a◦b)⇂q = a⇂qϑ−1
+b ◦b⇂q
+for all p ≤ d(a) and q ≤ r(b). (2.13)
+This can be extended to restrictions of compositions with an arbitrary number of terms. In a
+sense, the first part of the next result is a formalisation of the previous observation.
+Lemma 2.14. Let C be an ordered ∗-category, and let a, b ∈ C and p ∈ vC.
+(i) If r(a) = d(b), then ϑa◦b = ϑaϑb.
+(ii) ϑp = idp↓.
+(iii) If p ≤ d(a), then ϑp⇃a = ϑa|p↓.
+Proof. (i). Write p = d(a), q = r(a) = d(b) and r = r(b). Note that
+ϑa : p↓ → q↓,
+ϑb : q↓ → r↓
+and
+ϑa◦b : p↓ → r↓,
+so that ϑa◦b and ϑaϑb are defined on the same domain, i.e. on p↓. Now let s ≤ p, and put
+t = sϑa = r(s⇃a). Then by (O5)′ we have
+sϑa◦b = r(s⇃(a ◦ b)) = r(s⇃a ◦ t⇃b) = r(t⇃b) = tϑb = sϑaϑb.
+(ii). By part (i), we have ϑp = ϑp◦p = ϑpϑp, and the result follows since the only idempotent
+bijection p↓ → p↓ is the identity.
+(iii). Both maps have domain p↓, so it suffices to show that tϑp⇃a = tϑa for all t ≤ p. For this
+we use (O4)′ to calculate
+tϑp⇃a = r(t⇃p⇃a) = r(t⇃a) = tϑa.
+We will also need the following simple property of the ϑ maps:
+Lemma 2.15. If C is an ordered ∗-category, and if a ∈ C, then ϑa is order-preserving, in the
+sense that
+p ≤ q
+⇒
+pϑa ≤ qϑa
+for all p, q ∈ d(a)↓.
+Proof. If p ≤ q ≤ d(a), then p⇃a = p⇃q⇃a ≤ q⇃a, and so r(p⇃a) ≤ r(q⇃a), i.e. pϑa ≤ qϑa.
+14
+
+In later chapters we will define v-congruences by specifying sets of generating pairs:
+Definition 2.16. Suppose C is a category, and Ω ⊆ C × C a set of pairs satisfying
+• (u, v) ∈ Ω ⇒ [d(u) = d(v) and r(u) = r(v)].
+The (v-)congruence generated by Ω, denoted Ω♯, is the least congruence on C containing Ω.
+Specifically, we have (a, b) ∈ Ω♯ if and only if there is a sequence
+a = a1 → · · · → ak = b
+such that for each 1 ≤ i < k we have
+ai = bi ◦ ui ◦ ci
+and
+ai+1 = bi ◦ vi ◦ ci
+for some bi, ci ∈ C and (ui, vi) ∈ Ω ∪ Ω−1.
+(Here Ω−1 = {(v, u) : (u, v) ∈ Ω}.)
+The next result shows how conditions (C4) and (C5) regarding a congruence ≈ = Ω♯ can be
+deduced from properties of its generating set Ω. In the case of (C5) we utilise the above maps ϑa.
+Lemma 2.17. Suppose ≈ = Ω♯ is a v-congruence on an ordered ∗-category C.
+(i) If Ω satisfies the condition
+(a, b) ∈ Ω
+⇒
+a∗ ≈ b∗,
+(2.18)
+then ≈ satisfies condition (C4).
+(ii) If Ω satisfies the condition
+(a, b) ∈ Ω
+⇒
+ϑa = ϑb
+and
+p⇃a ≈ p⇃b for all p ≤ d(a),
+(2.19)
+then ≈ satisfies condition (C5).
+Proof. We just prove the second part, as the first is easier. To show that (C5) holds, suppose
+a ≈ b, and let the ai, bi, ci, ui, vi be as in Definition 2.16. Of course it suffices to show that
+p⇃ai ≈ p⇃ai+1 for each i. For this we define
+q = pϑbi
+and
+r = qϑui.
+Since ϑui = ϑvi by (2.19), we also have r = qϑvi. We then use (O5)′ and (2.19) to calculate
+p⇃ai = p⇃(bi ◦ ui ◦ ci) = p⇃bi ◦ q⇃ui ◦ r⇃ci ≈ p⇃bi ◦ q⇃vi ◦ r⇃ci = p⇃(bi ◦ vi ◦ ci) = p⇃ai+1.
+2.3
+Regular ∗-semigroups
+As we have already mentioned, our primary interest is in the class of regular ∗-semigroups, as
+defined by Nordahl and Scheiblich in [114]. Here we revise their definition, and list some of their
+basic properties. The main new results will be proved later in the paper, but we will give a few
+‘appetisers’ here to motivate these later results.
+Definition 2.20. A regular ∗-semigroup is a semigroup S with a unary operation ∗ : S → S : s �→ s∗
+satisfying
+(a∗)∗ = a = aa∗a
+and
+(ab)∗ = b∗a∗
+for all a, b ∈ S.
+15
+
+Note that a regular ∗-semigroup S is considered to be an algebra of type (2, 1), i.e. ∗ is a basic
+operation of S. There are semigroups with multiple unary operations, giving rise to pairwise
+non-isomorphic regular ∗-semigroups [126].
+Any inverse semigroup is a regular ∗-semigroup,
+with the involution being inversion, a∗ = a−1. In fact, it has long been known [127] that inverse
+semigroups are precisely the regular ∗-semigroups satisfying the additional identity
+aa∗bb∗ = bb∗aa∗
+for all a, b ∈ S.
+(2.21)
+As we will see, inverse semigroups represent a somewhat ‘degenerate’ case in the theory we
+develop; see Chapter 9.
+Arguably the most important families of non-inverse regular ∗-semigroups are the so-called
+diagram monoids, including the Brauer monoids [10], Temperley-Lieb monoids [130], partition
+monoids [76,98], Motzkin monoids [6] and several others. In Section 2.4 we will look extensively
+at the partition monoids, but before this we give the following simple example, coming from a
+standard semigroup-theoretic construction; we will return to it on a number of occasions. Another
+standard generalisation of the construction is given in Example 5.24 (see also Example 5.21);
+more examples of this kind can also be found in [118].
+Example 2.22. Let P be an arbitrary set, and let S = P × P = {(p, q) : p, q ∈ P} be the
+cartesian product of two copies of P. We define binary and unary operations on S by:
+(p, q)(r, s) = (p, s)
+and
+(p, q)∗ = (q, p).
+It is then routine to check that S is a regular ∗-semigroup. This semigroup is known as the
+square band over P.
+(More generally, a rectangular band has the form P × Q, for possibly
+different sets P and Q, with the same product as above; this is still a semigroup, but need not
+have an involution.)
+For a regular ∗-semigroup S, we write
+E = E(S) = {e ∈ S : e2 = e}
+and
+P = P(S) = {p ∈ S : p2 = p = p∗}.
+So E consists of all idempotents of S. The elements of P are called projections, and of course
+we have P ⊆ E.
+Projections play a very important role in virtually every study of regular
+∗-semigroups in the literature, and this is also true in the current work. The following result
+gathers some of the basic properties of idempotents and projections. Proofs can be found in many
+places (e.g., [70]), but we give some simple arguments here to keep the paper self-contained.
+Lemma 2.23. If S is a regular ∗-semigroup, with sets of idempotents E = E(S) and projections
+P = P(S), then
+(i) P = {aa∗ : a ∈ S} = {a∗a : a ∈ S},
+(ii) E = P 2 = {pq : p, q ∈ P}, and consequently ⟨E⟩ = ⟨P⟩,
+(iii) a∗Pa ⊆ P for all a ∈ S.
+Proof. (i). It is enough to show that P = {aa∗ : a ∈ S}. If p ∈ P, then p = pp = pp∗.
+Conversely, if a ∈ S, then (aa∗)2 = aa∗aa∗ = aa∗, and (aa∗)∗ = (a∗)∗a∗ = aa∗.
+(ii). If e ∈ E, then e = ee∗e = e(ee)∗e = ee∗ · e∗e, with ee∗, e∗e ∈ P, by part (i). Conversely, for
+any p, q ∈ P, we have pq = pq(pq)∗pq = pqq∗p∗pq = pqqppq = pqpq = (pq)2.
+(iii). If p ∈ P, then a∗pa = a∗p∗pa = (pa)∗pa ∈ P by part (i).
+16
+
+Comparing (2.21) with Lemma 2.23(i), it follows that inverse semigroups are precisely the
+regular ∗-semigroups with commuting projections. In fact, it is easy to see that any idempotent of
+an inverse semigroup is a projection, so another equivalent condition for a regular ∗-semigroup S
+to be inverse is that E(S) = P(S).
+If p, q ∈ P, then it follows from Lemma 2.23(iii) that pqp = p∗qp ∈ P. Thus, for all p ∈ P,
+we have a well-defined map
+θp : P → P
+given by
+qθp = pqp
+for q ∈ P.
+(2.24)
+The next result summarises some of the key properties of these maps.
+Lemma 2.25. If S is a regular ∗-semigroup, then for any p, q ∈ P we have
+pθp = p,
+θpθp = θp,
+pθqθp = qθp,
+θpθqθp = θqθp
+and
+θpθqθpθq = θpθq.
+Proof. The first two follow from the fact that projections are idempotents. The third and fifth
+follow from the fact that the product of two projections is an idempotent (cf. Lemma 2.23(ii)).
+For the fourth, given any t ∈ P we have
+tθqθp = tθpqp = pqp · t · pqp = tθpθqθp.
+When S is inverse, commutativity of projections implies that qθp = pθq = pq = qp for
+all p, q ∈ P. This then leads to a significant simplification in the theory developed below, as we
+will explore in detail in Chapter 9.
+Next we define a relation ≤ on P = P(S) by
+p ≤ q
+⇔
+p = pθq = qpq
+for p, q ∈ P.
+(2.26)
+Note that p ≤ q precisely when q is a left and right identity for p. Using this it is easy to see
+that ≤ is a partial order. It is also possible to deduce this fact from the identities listed in
+Lemma 2.25, as we do in Lemma 3.7 below. In fact, since p = qpq ⇔ p = pq = qp, and since
+p = pq ⇒ p = p∗ = (pq)∗ = q∗p∗ = qp,
+(2.27)
+and conversely by symmetry, it follows that
+p ≤ q
+⇔
+p = pq
+⇔
+p = qp.
+(2.28)
+The above properties of projections, and their associated θ mappings, will be of fundamen-
+tal importance in all that follows.
+In particular, even though P = P(S) is generally not a
+subsemigroup of the regular ∗-semigroup S, we can regard it as a unary algebra, whose basic
+(unary) operations are the θp (p ∈ P). In Chapter 3, we take the properties from Lemma 2.25
+as the axioms for what we will call a projection algebra (see Definition 3.1). Ultimately, we will
+see that (abstract) projection algebras are precisely the unary algebras of projections of regular
+∗-semigroups. This has in fact been known for some time (see for example [72]), but our new
+approach leads to another way to see this. When S is inverse, P is simply a subsemigroup of S,
+and there is no real advantage in considering P as a unary algebra. In a sense, a projection
+algebra is the ‘∗-analogue’ of the semilattice of idempotents of an inverse semigroup.
+As a foreshadowing of things to come, we explain now how to use the projections of a regular
+∗-semigroup S to construct a groupoid G = G(S). The groupoid G has the same underlying set
+as S, and the (partially defined) composition in G is a restriction of the (totally defined) product
+in S. Roughly speaking, G remembers only the ‘nice’ or ‘easy’ products; see Section 2.4 for some
+examples justifying our use of these words.
+17
+
+As it happens, the construction of G = G(S) is exactly the same as in the inverse case; see
+[87, Chapter 4]. However, we will see that the groupoid G(S) is not a total invariant of the
+regular ∗-semigroup S, unlike the case for inverse semigroups; see Section 5.2 for more details.
+The definition of G = G(S) begins with the following neat equational (rather than existential)
+characterisation of Green’s relations on a regular ∗-semigroup.
+Lemma 2.29. If S is a regular ∗-semigroup, and if a, b ∈ S, then
+a R b ⇔ aa∗ = bb∗
+and
+a L b ⇔ a∗a = b∗b.
+Proof. Even though this has been proved in a number of places, we prove the statement con-
+cerning R to keep the paper self contained. The statement for L is of course dual.
+Suppose first that a R b, so that a = bx for some x ∈ S. Using (2.28), it then follows that
+aa∗ = bxa∗ = bb∗bxa∗ = bb∗aa∗
+⇒
+aa∗ ≤ bb∗.
+By symmetry, we also have bb∗ ≤ aa∗, so that aa∗ = bb∗.
+Conversely, if aa∗ = bb∗, then a = aa∗a = b(b∗a), and similarly b = a(a∗b), so that a R b.
+For a regular ∗-semigroup S, and for a ∈ S, we write
+d(a) = aa∗
+and
+r(a) = a∗a.
+Both of these elements are projections (cf. Lemma 2.23(i)), and the identity a = aa∗a gives
+a = d(a) · a = a · r(a).
+Moreover, Lemma 2.29 says that
+a R b ⇔ d(a) = d(b)
+and
+a L b ⇔ r(a) = r(b).
+Since p = pp∗ = p∗p for any projection p, we have d(p) = r(p) = p for such p. It then follows
+from Lemma 2.29 that the R- and L -classes of p are given by
+Rp = {a ∈ S : aa∗ = pp∗} = {a ∈ S : d(a) = p}
+and similarly
+Lp = {a ∈ S : r(a) = p}.
+(2.30)
+Lemma 2.31. If S is a regular ∗-semigroup, and if a, b ∈ S are such that r(a) = d(b), then
+d(ab) = d(a)
+and
+r(ab) = r(b).
+Proof. We just prove the first claim, as the second is symmetrical. If r(a) = d(b), then a∗a = bb∗,
+and then
+d(ab) = ab(ab)∗ = abb∗a = aa∗aa∗ = aa∗ = d(a).
+This allows us to make the following definition.
+Definition 2.32. Given a regular ∗-semigroup S, we define the category G = G(S) as follows.
+• The object set of G is vG = P = P(S).
+• For a ∈ G we have d(a) = aa∗ and r(a) = a∗a.
+• For a, b ∈ G with r(a) = d(b), we have a ◦ b = ab.
+18
+
+For projections p, q ∈ P, it follows from (2.30) that
+G(p, q) = {a ∈ S : d(a) = p, r(a) = q} = Rp ∩ Lq.
+Since D = R ◦ L , such a morphism set is non-empty precisely when p D q. It follows that the
+connected components of G = G(S) are in one-one correspondence with the D-classes of S.
+Since the underlying set of G = G(S) is S, the unary operation ∗ : S → S can be thought of
+as a map ∗ : G → G. It is a routine matter to verify conditions (I1)–(I4) from Definition 2.1, so
+we have the following:
+Proposition 2.33. If S is a regular ∗-semigroup, then the category G = G(S) is a groupoid, with
+inversion given by −1 = ∗.
+Remark 2.34. We will see later that the groupoid G = G(S) associated to a regular ∗-
+semigroup S has rather a lot more structure than is evident at this stage. For example, the
+fact that the object set vG = P is a (so-called) projection algebra will allow us to construct an
+order on G. This ordering can then be used to reduce the calculation of an arbitrary product ab
+in S to an associated composition in G:
+ab = a′ ◦ e ◦ b′,
+(2.35)
+where a′ ≤ a and b′ ≤ b, and where e ∈ G is a special morphism r(a′) → d(b′). Although we do
+not need to understand the ordering on G yet, we can at least verify that a composition of the
+form (2.35) does indeed exist. Specifically, let us write
+p = r(a) = a∗a
+and
+q = d(b) = bb∗,
+noting that a = ap and b = qb. We also define the additional projections
+p′ = qθp = pqp
+and
+q′ = pθq = qpq.
+Then since pq is an idempotent, by Lemma 2.23(ii), we have p′q′ = (pq)3 = pq, and so
+ab = ap · qb = a · p′q′ · b = ap′ · p′q′ · q′b.
+Then with a′ = ap′, e = p′q′(= pq) and b′ = q′b, one can check that
+r(a′) = p′ = d(e)
+and
+r(e) = q′ = d(b′).
+For example,
+r(a′) = (ap′)∗ap′ = (apqp)∗apqp = pqpa∗apqp = pqp · p · pqp = pqpqp = pqp = p′.
+Thus, continuing from above, it follows that indeed
+ab = ap′ · p′q′ · q′b = a′ · e · b′ = a′ ◦ e ◦ b′.
+In what follows, we will think of a′ = ap′ and b′ = q′b as ‘restrictions’ a′ = a⇂p′ and b′ = q′⇃b in
+the groupoid G.
+Remark 2.36. There is a rather subtle point regarding Remark 2.34 that is far from obvious at
+this stage, but ought to be mentioned now. It may seem as if we are implying that the structure
+of the regular ∗-semigroup S is completely determined by that of the groupoid G = G(S). That
+is, if we are given a complete description of the groupoid G, including its composition, inversion,
+ordering, and the (so-called) projection algebra structure of its object set P = vG, then it might
+seem that we ought to be able to construct the entire multiplication table of S. However, this
+is far from the truth. Indeed, in Section 5.2, we will give examples of non-isomorphic regular
+19
+
+∗-semigroups S1 and S2 with exactly the same groupoids G(S1) = G(S2). (We postpone the
+definition of these semigroups, as we have already gone on a somewhat lengthy tangent, and
+also since we will only be able to fully appreciate their properties once we have developed more
+theory.)
+If the reader is worried that this contradicts the previous remark, the source of the subtlety
+is in the precise identity of the element e from (2.35) that allowed us to reduce a product ab in S
+to a composition in G:
+ab = a⇂p′ ◦ e ◦ q′⇃b.
+(On the other hand, the projections p′, q′ can be found directly using the projection algebra
+structure of P = vG.) We were able to ‘locate’ this element e in Remark 2.34 becase we began
+with full knowledge of the semigroup S; we simply took e = pq = p′q′. However, if we begin
+with an ordered groupoid G, it is not so obvious what this element e should be, even if we know
+that G = G(S) for some (unknown) regular ∗-semigroup S. Certainly e should be a morphism
+p′ → q′, but when p′ ̸= q′, it is not so easy to distinguish any such morphism. Getting around this
+problem is one of the main sources of difficulty encountered in the current work. Our eventual
+solution utilises what we will call the ‘chain groupoid’ C = C (P) associated to an (abstract)
+projection algebra P, and a certain ‘evaluation map’ ε : C → G. We will find our element e in
+the image of this map.
+Remark 2.37. Before moving on, it is worth commenting on another related notion, namely
+the so-called trace of a semigroup. This goes back to a classical result of Miller and Clifford.
+Among other things, [106, Theorem 3] says that for elements a, b of a semigroup S, we have
+ab ∈ Ra ∩ Lb
+⇔
+La ∩ Rb contains an idempotent.
+Such products ab ∈ Ra ∩ Lb are often called trace products in the literature. We can then define
+a partial binary operation ⊙ on S by
+a ⊙ b =
+�
+ab
+if La ∩ Rb contains an idempotent
+undefined
+otherwise,
+and the resulting partial algebra (S, ⊙) is often called the trace of S. Following Definition 2.32,
+it is not hard to see that in a regular ∗-semigroup S, a composition a ◦ b exists precisely when
+La ∩ Rb contains a projection, which must of course be a∗a = bb∗, i.e. r(a) = d(b).
+Since
+projections are idempotents, it follows that a ◦ b being defined forces a ⊙ b to be defined, and
+then of course a ◦ b = a ⊙ b = ab, meaning that ◦ ⊆ ⊙, i.e. that ⊙ is an extension of ◦. We have
+already noted that S is inverse if and only if P(S) = E(S), and it now quickly follows that this
+is also equivalent to having ◦ = ⊙. In this inverse case, the trace (S, ⊙) is then precisely the
+groupoid G(S) from Definition 2.32, but this is not true of arbitrary regular ∗-semigroups.
+As we have already mentioned, projections have played a crucial role in virtually all studies
+of regular ∗-semigroups.
+Of particular significance in papers such as [22, 33, 39] are pairs of
+projections p, q ∈ P = P(S) that are mutual inverses, i.e. pairs that satisfy p = pqp and q = qpq.
+This is very much the case in the current work: so much so, in fact, that we define relations ≤F
+and F on P by
+p ≤F q
+⇔
+p = pqp = qθp
+and
+p F q
+⇔
+[p ≤F q and q ≤F p].
+(2.38)
+We think of pairs of F-related projections as friends (hence the symbol F). It is clear that ≤F
+and F are both reflexive, and that F is symmetric. But neither ≤F nor F is transitive in
+general (and neither is friendship in ‘real life’); for an example, see Section 2.4. It is easy to
+see that p ≤ q
+⇒
+p ≤F q, where ≤ is the partial order in (2.26). It follows quickly from
+Lemma 2.29 that
+p F q
+⇔
+p R pq L q
+⇔
+p L qp R q.
+(2.39)
+20
+
+It of course follows from this that
+p F q
+⇒
+p D q,
+(2.40)
+though the converse does not hold in general; again see Section 2.4. Combining (2.39) with
+[68, Proposition 2.3.7], it follows that
+p F q
+⇔
+Lp ∩ Rq contains an idempotent
+⇔
+Rp ∩ Lq contains an idempotent,
+and of course these idempotents are
+qp ∈ Lp ∩ Rq
+and
+pq ∈ Rp ∩ Lq.
+This all means that a pair of F-related projections p, q ∈ P(S) generates a 2 × 2 rectangular
+band in S:
+p
+pq
+qp
+q
+One can define a graph Γ(S) with vertex set P = P(S), and with (undirected) edges {p, q}
+whenever p ̸= q and p F q. Connectivity properties of (certain subgraphs of) these graphs
+played a vital role in [39], in studies of idempotent-generated regular ∗-semigroups and their
+ideals. We will say more about this in Section 2.4.
+The (standard) proof of Lemma 2.23(ii) given above involved showing that any idempotent
+e ∈ E = E(S) of a regular ∗-semigroup S satisfies e = ee∗ · e∗e, with ee∗, e∗e ∈ P = P(S). It
+quickly follows that the idempotents e, e∗ ∈ E(S) also generate a 2 × 2 rectangular band in S:
+e
+ee∗
+e∗e
+e∗
+This is in fact a characterisation of the biordered sets arising from regular ∗-semigroups, in a
+sense made precise in [113, Corollary 2.7]. In any case, it quickly follows that ee∗ F e∗e for
+any e ∈ E, so this shows that any idempotent is a product of two F-related projections, and it
+follows that not only do we have E = P 2 (cf. Lemma 2.23(ii)), but in fact
+E = {pq : (p, q) ∈ F}.
+Actually, such factorisations for idempotents are unique. Indeed, if e = pq, where e ∈ E and
+(p, q) ∈ F, then
+ee∗ = pq(pq)∗ = pqq∗p∗ = pqqp = pqp = p
+and similarly
+e∗e = q.
+(2.41)
+Consequently, we have |E| = |F|. A number of studies have enumerated the idempotents in
+various classes of regular ∗-semigroups [20,21,84]. The identity |E| = |F| played an implicit role
+in [21].
+So idempotents are products of F-related projections.
+As an application of the theory
+developed in this paper, we will be able to prove the following result, which extends this to
+arbitrary products of idempotents, though uniqueness is lost, in general (cf. (2.48)); see Proposi-
+tion 4.39(iii) and Remark 4.40. The result might be known, but we are not aware of its existence
+in the literature; it does bear some resemblance, however, to a classical result of FitzGerald [48].
+Moreover, we believe it could be useful in studies of idempotent-generated regular ∗-semigroups,
+and might have even led to some simpler arguments in existing studies such as [22,38,39].
+21
+
+Proposition 2.42. Let S be a regular ∗-semigroup, with E = E(S) and P = P(S). Then for
+any a ∈ ⟨E⟩ = ⟨P⟩ we have
+a = p1 · · · pk
+for some p1, . . . , pk ∈ P with p1 F · · · F pk.
+We will leave our preliminary discussion of regular ∗-semigroups here. We will return to them
+again in Chapter 5. But for now, we will discuss an important special case.
+2.4
+Case study: diagram monoids
+The theory of regular ∗-semigroups has seen something of a resurgence in recent years, partly due
+to the importance of so-called diagram monoids. Here we recall the definition of these monoids,
+and use them to illustrate some of the ideas discussed in the previous section. We will also return
+to them at various points during the rest of the paper. Here we focus exclusively on the partition
+monoids [76,98], although similar comments could be made for other diagram monoids, such as
+(partial) Brauer, Temperley-Lieb and Motzkin monoids [6,10,22,39,100,130].
+Let X be a set, and let X′ = {x′ : x ∈ X} be a disjoint copy of X. The partition monoid
+over X, denoted PX, is defined as follows. The elements of PX are the set partitions of X ∪ X′,
+and the operation will be defined shortly. A partition from PX will be identified with any graph
+on vertex set X ∪ X′ whose connected components are the blocks of the partition. The elements
+of X and X′ are called upper and lower vertices, respectively, and are generally displayed in two
+parallel rows, as in the various figures below.
+To define the product on PX, consider two partitions α, β ∈ PX. Let X′′ = {x′′ : x ∈ X} be
+another disjoint copy of X, and define three new graphs:
+• α∨, the graph on vertex set X ∪ X′′ obtained from α by changing each lower vertex x′ to x′′,
+• β∧, the graph on vertex set X′′ ∪ X′ obtained from β by changing each upper vertex x to x′′,
+• Π(α, β), the graph on vertex set X ∪ X′′ ∪ X′, whose edge set is the union of the edge sets
+of α∨ and β∧.
+The graph Π(α, β) is called the product graph of α and β; it is generally drawn with X′′ as the
+middle row of vertices. The product αβ is defined to be the partition of X ∪X′ with the property
+that u, v ∈ X ∪ X′ belong to the same block of αβ if and only if u, v are connected by a path
+in Π(α, β).
+If X = {1, . . . , n} for a non-negative integer n, we write Pn = PX.
+Example 2.43. To illustrate the product above, consider the partitions α, β ∈ P6 defined by
+α =
+�
+{1, 2, 3, 1′}, {4, 4′, 5′, 6′}, {5}, {6}, {2′, 3′}
+�
+,
+β =
+�
+{1, 4′, 6′}, {2, 3}, {4, 5, 6, 1′, 2′, 3′}, {5′}
+�
+.
+Figure 1 illustrates (graphs representing) α and β, their product
+αβ =
+�
+{1, 2, 3, 4′, 6′}, {4, 1′, 2′, 3′}, {5}, {6}, {5′}
+�
+,
+as well as the product graph Π(α, β). Here and elsewhere, vertices are assumed to increase from
+left to right, 1, . . . , n, and similarly for (double) dashed vertices.
+A reader familiar with partition monoids might protest that we have ‘cheated’ in Exam-
+ple 2.43, and that our choice of α and β was too ‘easy’ to fully illustrate the complexities of
+calculating products of partitions. This is indeed the case, as the bottom half of α ‘matches’ the
+top half of β in a way that can hopefully be understood by examining Figure 1, but which will
+be made precise shortly. Before this, we briefly consider a more ‘difficult’ product:
+22
+
+α =
+β =
+= αβ
+Figure 1. Left to right: partitions α, β ∈ P6, the product graph Π(α, β), and the product αβ ∈ P6.
+For more information, see Example 2.43.
+Example 2.44. Figure 2 gives another product, this time with α, β ∈ P20. One can immediately
+see that there is no such ‘matching’ between the bottom of α and the top of β. Rather, to calculate
+the product αβ, one needs to follow paths in the product graph Π(α, β), often alternating several
+times between edges coming from α or β. For example, to see that {1′, 4′} is a block of αβ, one
+needs to trace the following path (or its reverse):
+1′
+β
+−−→ 1′′
+α
+−−→ 2′′
+β
+−−→ 3′′
+α
+−−→ 4′′
+β
+−−→ 4′.
+α =
+Π(α, β)
+β =
+αβ =
+Figure 2.
+Top: partitions α, β ∈ P20, already connected to create the product graph Π(α, β).
+Bottom: the product αβ ∈ P20. For more information, see Example 2.44.
+We hope that Example 2.44 convinces the reader that products in PX can be ‘messy’. Of
+course things get worse when we increase the number of vertices, even more so when X is
+infinite [35]. Nevertheless, it is actually quite easy to see that PX is a regular ∗-semigroup. The
+involution
+∗ : PX → PX : α �→ α∗
+can be defined diagrammatically as a reflection in a horizontal axis; see Figure 3. Formally, α∗
+is obtained from α by swapping dashed and undashed vertices, x ↔ x′. It is not hard to see that
+(α∗)∗ = α = αα∗α
+and
+(αβ)∗ = β∗α∗
+for all α, β ∈ PX.
+α =
+= α∗
+Figure 3. A partition α ∈ P6 (left) and its image α∗ under the involution of P6.
+23
+
+At this point it is convenient to recall some further notation and terminology. First, we say
+a non-empty subset ∅ ̸= A ⊆ X ∪ X′ is:
+• a transversal if A ∩ X ̸= ∅ and A ∩ X′ ̸= ∅ (i.e. if A contains both upper and lower vertices),
+• an upper non-transversal if A ⊆ X (i.e. if A contains only upper vertices), or
+• a lower non-transversal if A ⊆ X′ (i.e. if A contains only lower vertices).
+We can then describe a partition α ∈ PX using a convenient two-line notation from [38]. Specif-
+ically, we write
+α =
+�Ai Cj
+Bi Dk
+�
+i∈I, j∈J, k∈K
+to indicate that α has transversals Ai ∪ B′
+i (i ∈ I), upper non-transversals Cj (j ∈ J) and
+lower non-transversals Dk (k ∈ K). We often abbreviate this to α =
+�Ai Cj
+Bi Dk
+�
+, with the indexing
+sets I, J and K being implied, rather than explicitly listed. When X is finite we list the blocks
+of partitions, viz. α =
+�A1 . . . Aq C1 . . . Cs
+B1 . . . Bq D1 . . . Dt
+�
+. Thus, for example, the partition β ∈ P6 from Figure 1
+can be expressed as
+β =
+� 1
+4, 5, 6 2, 3
+4, 6 1, 2, 3
+5
+�
+.
+With this notation, the identity of PX is the partition idX =
+�
+x
+x
+�
+x∈X, and the involution is given
+by
+α =
+�Ai Cj
+Bi Dk
+�
+�→ α∗ =
+�Bi Dk
+Ai Cj
+�
+.
+Moreover, one can easily see that with α =
+�Ai Cj
+Bi Dk
+�
+, and with the notation of Section 2.3, we
+have
+d(α) = αα∗ =
+�Ai Cj
+Ai Cj
+�
+and
+r(α) = α∗α =
+�Bi Dk
+Bi Dk
+�
+.
+Thus, consulting Lemma 2.29, it follows that partitions α, β ∈ PX are R-related (or L -related)
+precisely when the ‘top halves’ (or ‘bottom halves’) of α, β ‘match’ in a sense that we again hope
+is clear. Nevertheless, this ‘matching’ can be formalised by defining some further notation.
+For α ∈ PX, we define the (co)domain and (co)kernel of α by:
+dom(α) = {x ∈ X : x belongs to a transversal of α},
+codom(α) = {x ∈ X : x′ belongs to a transversal of α},
+ker(α) = {(x, y) ∈ X × X : x and y belong to the same block of α},
+coker(α) = {(x, y) ∈ X × X : x′ and y′ belong to the same block of α}.
+The rank of α, denoted rank(α), is the number of transversals of α. Thus, dom(α) and codom(α)
+are subsets of X, ker(α) and coker(α) are equivalences on X, and rank(α) is a cardinal between 0
+and |X|. For example, with α ∈ P6 from Figure 1, we have
+dom(α) = {1, 2, 3, 4},
+ker(α) = (1, 2, 3 | 4 | 5 | 6),
+codom(α) = {1, 4, 5, 6},
+coker(α) = (1 | 2, 3 | 4, 5, 6),
+rank(α) = 2,
+where we have indicated equivalences by listing their classes. The various parts of the following
+result are contained in [42,51,137], though some of those papers use different terminology.
+24
+
+Lemma 2.45. For α, β ∈ PX, we have
+(i) α R β ⇔ [dom(α) = dom(β) and ker(α) = ker(β)],
+(ii) α L β ⇔ [codom(α) = codom(β) and coker(α) = coker(β)],
+(iii) α D β ⇔ α J β ⇔ rank(α) = rank(β).
+The D = J -classes of PX are the sets
+Dµ = Dµ(PX) = {α ∈ PX : rank(α) = µ}
+for cardinals 0 ≤ µ ≤ |X|.
+Group H -classes contained in Dµ are isomorphic to the symmetric group Sµ.
+Let us now return to the example products considered earlier.
+Example 2.46. Consider again the partitions α, β ∈ P6 from Example 2.43 and Figure 1. The
+projections
+r(α) = α∗α
+and
+d(β) = ββ∗
+are calculated in Figure 4. Since we have r(α) = d(β), we see that α and β are composable in
+the groupoid G(P6) from Definition 2.32, so that in fact αβ = α◦β. This explains the ‘matching’
+phenomenon discussed above, and gives the reason for the ‘ease’ of forming the product αβ.
+α∗ =
+α =
+r(α) =
+β =
+β∗ =
+d(β) =
+Figure 4. The projections r(α) = α∗α and d(β) = ββ∗, where α, β ∈ P6 are as in Figure 1. For
+more information, see Examples 2.43 and 2.46.
+Example 2.47. On the other hand, the partitions α, β ∈ P20 from Example 2.44 and Figure 2
+are not composable in the groupoid G = G(P20). Indeed, one can check that the projections
+r(α) = α∗α
+and
+d(β) = ββ∗
+are as shown in Figure 5, and we clearly do not have r(α) = d(β). On the other hand, it will
+follow from results of later chapters that there exist certain partitions α′ ≤ α and β′ ≤ β (the ≤
+relation will be defined later) that can ‘almost’ be composed in G. By this we mean that while
+we still do not have r(α′) = d(β′), we do have r(α′) F d(β′). (The F relation was defined
+in (2.38).) It will follow from the general theory that
+α = α′ ◦ ε ◦ β′,
+where ε ∈ G is a special element with d(ε) = r(α′) and r(ε) = d(β′). The partitions α′, ε, β′ ∈ P20
+are shown in Figure 6. The reader might like to check that ε is an idempotent of P20, i.e. that ε2 = ε
+(but we note that the composition ε ◦ ε does not exist in G).
+25
+
+r(α) =
+d(β) =
+Figure 5. The projections r(α) = α∗α and d(β) = ββ∗, where α, β ∈ P20 are as in Figure 2. For
+more information, see Examples 2.44 and 2.47.
+ε =
+α′ =
+Π(α′, ε, β′)
+β′ =
+α′ ◦ ε ◦ β′ =
+Figure 6. Top: partitions α′, ε, β′ ∈ P20, with the edges of ε shown in red. Bottom: the composition
+α′ ◦ ε ◦ β′ in the groupoid G(P20). Note that α′ ◦ ε ◦ β′ = αβ, where α, β ∈ P20 are as in Figure 2.
+For more information, see Examples 2.44 and 2.47.
+We conclude this section with a brief discussion of the relations ≤, ≤F and F on the set
+P = P(PX) of projections of a partition monoid PX. These relations were defined in (2.26)
+and (2.38). First, one can check that the projections of the partition monoid P2 are the six
+partitions shown in Figure 7.
+Figure 8 shows the partial order ≤ on P(P2), as defined in (2.26); as usual for Hasse dia-
+grams, only covering relationships are shown, and the rest can be deduced from transitivity (and
+reflexivity). Figure 9 shows the relations ≤F and F on P(P2), as defined in (2.38), and we
+remind the reader that these are not transitive. For example, we have
+F
+F
+,
+even though the first and third of the above projections are not F-related, or even ≤F-related.
+In both Figures 8 and 9, the elements of P = P(P2) have been arranged so that each row
+consists of D-related elements; cf. Lemma 2.45. So from top to bottom, the rows correspond to
+the projections from the D-classes D2, D1 and D0, where
+Di = Di(P2) = {α ∈ P2 : rank(α) = i}
+for i = 0, 1, 2.
+26
+
+Figure 7. The projections of the partition monoid P2.
+Figure 8. Hasse diagram of the poset P(P2), with respect to the order ≤ given in (2.26). An arrow
+p → q means p ≤ q.
+For i = 0, 1, 2, we write Pi = P ∩ Di for the set of projections from Di. The right-hand graph in
+Figure 9 is the graph Γ(P2), as defined at the end of Section 2.3. We denote this by Γ = Γ(P2);
+so the vertex set of Γ is P = P(P2), and Γ has an undirected edge {p, q} precisely when p ̸= q
+and p F q. As in (2.40), F-related projections are D-related. Thus,
+Γ = Γ0 ⊔ Γ1 ⊔ Γ2
+decomposes as the disjoint union of the induced subgraphs Γi on the vertex sets Pi, for i = 0, 1, 2.
+It is clear that each Γi is connected, but this is not necessarily the case for an arbitrary regular
+∗-semigroup S.
+In general, for a regular ∗-semigroup S, we still have the graph Γ(S), and we still have the
+decomposition
+Γ(S) =
+�
+D∈S/D
+Γ(D),
+where Γ(D) is the induced subgraph of Γ(S) on vertex set P(S) ∩ D, for each D-class D of S.
+However, the subgraphs Γ(D) need not be connected in general. For example, if S is inverse, then
+the F-relation is trivial; it follows that each Γ(D) is discrete (has empty edge set), and hence is
+disconnected if D contains more than one idempotent. On the other hand, it follows from results
+of [39] that the graphs Γ(D) are connected for any D-class D of a finite partition monoid Pn,
+and this turns out to be equivalent to certain facts about minimal idempotent-generation of the
+proper ideals of Pn. We have seen this connectivity property in the case n = 2, above. Figure 10
+shows the graph Γ(P3), produced using the Semigroups package for GAP [52, 107]. From left
+to right, the connected components of the graph Γ(P3) in Figure 10 are the induced subgraphs
+Γ(D3), Γ(D2), Γ(D1) and Γ(D0).
+27
+
+Figure 9. Left: the relation ≤F on P(P2); an arrow p → q means p ≤F q. Right: the relation F
+on P(P2); an edge p − q means p F q. These relations are given in (2.38). In both diagrams, loops
+are omitted.
+Figure 10. The graph Γ(P3), produced by GAP.
+As a special case, the induced subgraph Γ(Dn−1) of the graph Γ(Pn) corresponding to the
+D-class Dn−1 = Dn−1(Pn) has a very interesting structure. First, one can easily check that the
+projections of Pn contained in Dn−1 have the form
+· · ·
+· · ·
+· · ·
+· · ·
+1
+i
+n
+πi =
+· · ·
+· · ·
+· · ·
+· · ·
+· · ·
+· · ·
+1
+j
+k
+n
+and
+πjk =
+for each 1 ≤ i ≤ n and 1 ≤ j < k ≤ n. It is also easy to check that the only non-identical
+F-relations among these projections are
+πi F πij F πj.
+The graph Γ(Dn−1) is shown in Figure 11 in the case n = 5. In the figure, vertices representing
+projections πi or πjk are labelled simply with the subscripts i or jk.
+28
+
+1
+2
+3
+4
+5
+12
+13
+14
+15
+23
+24
+25
+34
+35
+45
+Figure 11. The graph Γ(D4), where D4 = D4(P5).
+The idempotent-generated subsemigroup E(PX) = ⟨E(PX)⟩ = ⟨P(PX)⟩ of a partition monoid PX
+was described for finite and infinite X in [34] and [38], respectively. One of the main results of [34]
+is that finite E(Pn) is (minimally) generated as a monoid by the set
+Ω = {πi, πjk : 1 ≤ i ≤ n, 1 ≤ j < k ≤ n}
+of all projections of rank n − 1, and that in fact
+E(Pn) = {idn} ∪ Sing(Pn),
+where here Sing(Pn) = Pn \ Sn is the singular ideal of Pn, consisting of all non-invertible ele-
+ments. The second main result was a presentation for Sing(Pn) in terms of the above generating
+set Ω. For example, one of the defining relations was
+πiπijπjπjkπkπkiπi = πiπikπkπkjπjπjiπi
+for distinct 1 ≤ i, j, k ≤ n,
+(2.48)
+where here we use symmetrical notation πst = πts. Examining Figure 11, one can see that the
+two words in this equation represent two different triangular paths in the graph Γ(Dn−1):
+i ⇝ j ⇝ k ⇝ i
+and
+i ⇝ k ⇝ j ⇝ i.
+(In representing the above paths we have omitted the intermediate vertices, so i ⇝ j is shorthand
+for i → ij → j, and so on.) Since paths in this graph correspond to lists of sequentially F-related
+projections, the above words represent factorisations of the form described in Proposition 2.42,
+and the corresponding tuples (πi, πij, πj, . . .) and (πi, πik, πk, . . .) are examples of what we will
+later call P-paths; see Section 3.2.
+The two factorisations represent the following partition,
+pictured in the case i < j < k:
+· · ·
+· · ·
+· · ·
+· · ·
+· · ·
+· · ·
+· · ·
+· · ·
+1
+i
+j
+k
+n
+Much more could be said, but we will conclude our preliminary discussion of partition monoids
+here. We will return to them at various stages throughout the text.
+29
+
+Part I
+Structure
+This first part of the paper is devoted to our main structural result, Theorem 6.7. This theorem
+states that the category of regular ∗-semigroups is isomorphic to the category of so-called chained
+projection groupoids. Such groupoids play the same role in the theory of regular ∗-semigroups
+that inductive groupoids play in inverse semigroup theory. Roughly speaking, we do not assume
+that the object sets of our groupoids are semilattices (as per inductive groupoids), but rather
+projection algebras. These are unary algebras that abstractly model the projections of regular
+∗-semigroups, together with their ‘conjugation actions’ as in (2.24). In fact, we will see that a
+groupoid simply having a projection algebra for its object set is not enough. Rather, our chained
+projection groupoids are pairs (G, ε), where G is a groupoid with vG = P a projection algebra,
+and where ε is a certain special functor that encapsulates the strong relationship between the
+groupoid structure of G and the algebra structure of P. The domain of the functor ε is another
+groupoid, C = C (P), which we call the chain groupoid of P. In a sense, C models the behaviour
+of P at the most ‘free’ level, somehow recording precisely the information that holds in every
+regular ∗-semigroup with projection algebra P. The full extent of this ‘free-ness’ will be explored
+in more detail in the second part of the paper; see Chapter 7.
+This part of the paper contains four chapters:
+• Chapter 3 introduces projection algebras, and their chain groupoids.
+• Chapter 4 defines chained projection groupoids, and shows how to associate a regular ∗-
+semigroup to each such groupoid; see Theorem 4.36.
+• Chapter 5 goes in the reverse direction, and shows how a regular ∗-semigroup gives rise to a
+chained projection groupoid; see Theorem 5.19.
+• Chapter 6 contains the main result, Theorem 6.7, which shows that the constructions of
+Chapters 4 and 5 are in fact mutually inverse isomorphisms between the categories of regular
+∗-semigroups and chained projection groupoids.
+The introduction of each chapter contains a detailed summary of its structure, and of the results
+it contains.
+3
+Projection algebras
+One of the key ideas in this paper is that of a projection algebra. Such an algebra consists of a
+set P, along with a family θp (p ∈ P) of unary operations, one for each element of P. These
+operations are required to satisfy certain axioms, as listed in Definition 3.1 below, and are meant
+to abstractly model the unary algebras of projections of regular ∗-semigroups. This is in fact
+not a new concept. Indeed, projection algebras have appeared in a number of settings, under a
+variety of names, including the P-groupoids of Imaoka [72], the P-sets of Yamada [138], (certain
+special) P-sets of Nambooripad and Pastijn [113], and the (left and right) projection algebras
+of Jones [73]. (We also mention Yamada’s p-systems [139]; these are defined in a very different
+way, but for a similar purpose. Strictly speaking, Yamada’s P-sets are different to Imaoka’s
+P-groupoids, but are ultimately equivalent.) We prefer Jones’ terminology ‘projection algebras’,
+since these are indeed algebras, not just sets, though we note that Jones considered binary
+algebras rather than Imaoka’s unary algebras, which we use here; see Remark 3.2. We also only
+use the term ‘groupoid’ in its usual categorical sense. Imaoka used it in one of its less-common
+30
+
+meanings, stemming from the fact that a projection algebra also has a partially defined binary
+operation; curiously this partial operation plays no role in the current work.
+Although the concept of a projection algebra is not new, here we build several new categorical
+structures on top of such algebras. The most general situation is covered in Chapter 4 below,
+where we have an ordered groupoid G whose object set vG = P is a projection algebra. The
+current chapter lays the foundation for this, by showing how to construct various categories
+directly from a projection algebra. The main such construction is the chain groupoid C = C (P);
+this will be used in the definition of the groupoids G in Chapter 4, but we will uncover its deeper
+categorical significance in Chapter 7.
+We begin in Section 3.1 by giving/recalling the definition of projection algebras (see Def-
+inition 3.1), and establishing some of their important properties. In Sections 3.2 and 3.4, re-
+spectively, we introduce the path category P = P(P) and the chain groupoid C = C (P) of a
+projection algebra P (see Definitions 3.17 and 3.39). The groupoid C is defined as a quotient
+of the category P by a certain congruence ≈, whose definition requires the somewhat-technical
+notion of linked pairs, which are the subject of Section 3.3.
+3.1
+Definitions and basic properties
+Here then is the key definition:
+Definition 3.1. A projection algebra is a set P, together with a collection of unary opera-
+tions θp (p ∈ P) satisfying the following axioms, for all p, q ∈ P:
+(P1) pθp = p,
+(P2) θpθp = θp,
+(P3) pθqθp = qθp,
+(P4) θpθqθp = θqθp,
+(P5) θpθqθpθq = θpθq.
+The elements of a projection algebra are called projections.
+Strictly speaking, one should refer to ‘a projection algebra (P, θ)’, but we will almost always
+use the symbol θ to denote the unary operations of such an algebra, and so typically refer to
+‘a projection algebra P’. On the small number of occasions we need to refer simultaneously to
+more than one projection algebra, we will be careful to distinguish the operations.
+Remark 3.2. It is worth noting at the outset that the collection of unary operations θp (p ∈ P)
+could be replaced by a single binary operation ⋄, defined by q ⋄p = qθp. For example, Jones took
+this binary approach in [73], in his study of the more general P-restriction semigroups, though
+his preferred symbol was ⋆, which we reserve for other uses throughout the paper. We prefer the
+unary approach of Imaoka [72], however, for three main reasons:
+• First, we feel that the meaning of (some of) axioms (P1)–(P5) is clearer in the unary context.
+For example, the binary form of (P4) is
+r ⋄ (q ⋄ p) = ((r ⋄ p) ⋄ q) ⋄ p
+for all p, q, r ∈ P.
+As we will discuss in more detail in Remark 4.9, axiom (P4) can be thought of as a rule for
+‘iterating’ the unary operations, and we feel that this intuition is somewhat lost in the binary
+form of (P4) above, although this is of course entirely subjective. (We also note that Jones’
+axioms are different to (P1)–(P5), though they are equivalent, as he explains in [73, Section 7].)
+31
+
+• The non-associativity of ⋄ means that brackets are necessary when working with ⋄-terms. On
+the other hand, associativity of function composition allows us to dispense with bracketing
+when using the θ maps, and this is a significant advantage when dealing with lengthy terms.
+• In later chapters, we will consider groupoids G whose object sets are projection algebras, vG = P.
+One of the key tools when studying such groupoids are certain maps
+Θa = θd(a)ϑa : P → P
+that are built from the unary operations of P and the maps ϑa : d(a)↓ → r(a)↓ defined
+in (2.10). These Θ maps could certainly be defined in terms of ⋄ and the ϑ maps, viz.
+pΘa = (p ⋄ d(a))ϑa,
+but their definition as a composition Θa = θd(a)ϑa is more direct, and leads to more suc-
+cinct statements and proofs, as for example with Proposition 4.6. The direct definition as a
+composition also helps to clarify the very formulation of our so-called (chained) projection
+groupoids; see Definitions 4.8 and 4.22.
+As we have already observed, the axioms (P1)–(P5) are abstractions of the properties of
+projections of regular ∗-semigroups; cf. Lemma 2.25.
+It will transpire (see especially Chap-
+ters 7 and 8) that abstract projection algebras are precisely the projection algebras of regular
+∗-semigroups, as is already known [72]. For the eager reader, Example 3.3 below provides a
+construction (without proof at this stage) of a regular ∗-semigroup with projection algebra P.
+In any case, the reader can keep in mind that (abstract) projection algebras are supposed to
+‘model’ the behaviour of algebras of projections of regular ∗-semigroups; thus, one can interpret
+every result in this chapter in the context of regular ∗-semigroups, and can gain an intuition for
+their meaning in that context.
+Example 3.3. Consider a projection algebra P. Since an operation θp is a map P → P, we
+can think of it as an element of the full transformation monoid TP . (This is the monoid of all
+maps P → P, under composition.) The maps θp generate the subsemigroup
+SP = ⟨θp : p ∈ P⟩ = {θp1 · · · θpk : k ≥ 1, p1, . . . , pk ∈ P} ≤ TP .
+So SP is an idempotent-generated semigroup (cf. (P2)), but it is generally not a regular ∗-
+semigroup. However, we can alter it to create a regular ∗-semigroup. Indeed, first we write T op
+P
+for the opposite semigroup to TP . So T op
+P
+has the same underlying set as TP , and its product ⋆
+is given by α ⋆ β = βα. We then consider the subsemigroup of the direct product TP × T op
+P
+generated by all pairs (θp, θp):
+FP = ⟨(θp, θp) : p ∈ P⟩ = {(θp1 · · · θpk, θpk · · · θp1) : k ≥ 1, p1, . . . , pk ∈ P} ≤ TP × T op
+P .
+It turns out that FP is a regular ∗-semigroup with projection algebra (isomorphic to) P. In
+fact, in Chapter 8 we will see that (up to isomorphism) FP is the unique idempotent-generated
+fundamental regular ∗-semigroup with projection algebra P; see Theorem 8.24 and Remark 8.25.
+Note that FP is a subdirect product of SP and Sop
+P .
+For the rest of this section we fix a projection algebra P. In what follows, for x, y ∈ P, we
+write x =1 y to indicate that x = y by an application of (P1), and similarly for =2, and so on.
+Lemma 3.4. For any p1, . . . , pk, q, r ∈ P we have
+θqθp1···θpk = θpk · · · θp1θqθp1 · · · θpk.
+Proof. This follows by iterating (P4).
+32
+
+A number of relations on P will play a crucial role in all that follows. The first of these is
+defined by
+p ≤ q
+⇔
+p = pθq.
+(3.5)
+By (P2), it quickly follows that
+p ≤ q
+⇔
+p = rθq
+for some r ∈ P.
+(3.6)
+Lemma 3.7. ≤ is a partial order on P.
+Proof. Reflexivity follows from (P1). For anti-symmetry, suppose p ≤ q and q ≤ p, so that
+p = pθq and q = qθp. Then
+p =1 pθp = pθqθp =3 qθp = q.
+For transitivity, suppose p ≤ q and q ≤ r, so that p = pθq and q = qθr. Then
+p = pθq = pθqθr =4 pθrθqθr.
+By (3.6), this implies p ≤ r.
+By (3.6), the image of θp : P → P is precisely
+im(θp) = p↓ = {q ∈ P : q ≤ p},
+(3.8)
+the down-set of p in the poset (P, ≤).
+Lemma 3.9. If p ≤ q, then θp = θpθq = θqθp.
+Proof. Since p = pθq, we have θp = θpθq =4 θqθpθq. The claim then quickly follows from (P2).
+An equally important role will be played by two further relations, ≤F and F, defined as
+follows. For p, q ∈ P, we say that
+p ≤F q
+⇔
+p = qθp.
+(3.10)
+By (P1), ≤F is reflexive, but it need not be transitive. We also define
+F = ≤F ∩ ≥F,
+(3.11)
+which is the largest symmetric (and reflexive) relation contained in ≤F. So
+p F q
+⇔
+p = qθp and q = pθq.
+Lemma 3.12. For any p, q ∈ P,
+(i) pθq ≤F p,
+(ii) p ≤ q ⇒ p ≤F q,
+(iii) p ≤F q ⇒ θp = θpθqθp.
+(iv) p F q ⇒ θp = θpθqθp and θq = θqθpθq.
+Proof. (i). We have pθpθq =4 pθqθpθq =3 qθpθq =3 pθq.
+(ii). If p ≤ q, then p = pθq, so qθp = qθpθq =4 qθqθpθq =1 qθpθq =3 pθq = p.
+(iii) and (iv). These follow immediately from (P4).
+33
+
+Remark 3.13. The partial order ≤ from (3.5) was used by Imaoka in [72], where it was defined
+by
+p ≤ q
+⇔
+p = pθq = qθp.
+Note that part (ii) of the lemma just proved means that the ‘= qθp’ part of Imaoka’s definition
+is superfluous. Jones also used the order ≤ in [73], defined exactly as in (3.5), albeit in binary
+form, p ≤ q ⇔ p = p ⋄ q (cf. Remark 3.2). We are not aware of any previous use of the ≤F
+relation in the literature, however, despite its central importance in the current work.
+The following simple result will be crucial in later chapters. Roughly speaking, it says that
+any pair of projections p, q ∈ P sit above an F-related pair p′, q′ ∈ P.
+Lemma 3.14. Let p, q ∈ P be arbitrary, and let p′ = qθp and q′ = pθq. Then
+p′ ≤ p,
+q′ ≤ q
+and
+p′ F q′.
+Proof. We obtain p′ ≤ p and q′ ≤ q directly from (3.6). We also have
+p′θq′ = qθpθpθq =4 qθpθqθpθq =5 qθpθq =3 pθq = q′
+and similarly
+q′θp′ = p′.
+Although ≤F is not transitive, ≤ and ≤F have some ‘transitivity-like’ properties in combi-
+nation:
+Lemma 3.15. If p, q, r ∈ P are such that p ≤ q ≤F r or p ≤F q ≤ r, then
+(i) p ≤F r,
+(ii) p F pθr.
+Proof. Throughout the proof we assume that p ≤ q ≤F r. The arguments for the case of
+p ≤F q ≤ r are similar, and are omitted.
+Since p ≤ q and q ≤F r, we have p = pθq and q = rθq. Since also p ≤F q by Lemma 3.12(ii),
+we also have p = qθp.
+(i). We must show that p = rθp. Since p ≤ q, Lemma 3.9 gives θp = θqθp. Combining all of the
+above gives rθp = rθqθp = qθp = p.
+(ii). We have pθr ≤F p by Lemma 3.12(i), so we just need to show that p ≤F pθr, i.e. that
+p = (pθr)θp. This follows from the axioms and the above-mentioned consequences of p ≤ q ≤F r:
+pθrθp =3 rθp = rθpθq =4 rθqθpθq = qθpθq =3 pθq = p.
+It remains an open problem to determine whether a projection algebra P (including its θ
+operations) can be somehow characterised by the purely order-theoretic properties of the relations
+≤ and ≤F. The next result is included in case it is of use in such a characterisation.
+Lemma 3.16. If p, q ∈ P are such that p ≤F q, then we have p F p′ ≤ q for some p′ ∈ P.
+Proof. It is a routine matter to verify that p′ = pθq has the stated properties.
+3.2
+The path category
+For the duration of this section we fix a projection algebra P, as in Definition 3.1, including the
+operations θp (p ∈ P), and the relations ≤, ≤F and F.
+Roughly speaking, we wish to provide an abstract setting in which we can think about ‘prod-
+ucts’ of projections p1 · · · pk, even though P itself does not have a binary operation. Guided by
+the (as-yet unproved) Proposition 2.42, we are (for now) solely interested in such ‘products’ in
+34
+
+the case that p1 F · · · F pk. The path category P (see Definition 3.17) is the first attempt
+at doing so; it represents such a ‘product’ as a tuple (p1, . . . , pk), which is considered as a mor-
+phism p1 → pk. In Section 3.4, we will define the chain groupoid C (see Definition 3.39) as a
+quotient C = P/≈ by a certain congruence ≈. The intuition here is that the chain groupoid
+records more information about such products than the factors alone, but only such informa-
+tion that is present in any regular ∗-semigroup with projection algebra P. For example, (p, p)
+and (p) should always represent the same ‘product’, since projections are idempotents; so too
+should (p, q, p) and (p) when p F q (cf. (2.38)). These pairs of paths (and one other family of
+pairs) are taken as generators for the congruence ≈. But before we get ahead of ourselves, here
+is the main definition for the current section:
+Definition 3.17. A P-path in a projection algebra P (cf. Definition 3.1) is a tuple
+p = (p1, p2, . . . , pk) ∈ P k
+for some k ≥ 1, such that p1 F p2 F · · · F pk.
+(Since F is not transitive, this does not imply that the pi are all F-related.) We say that p is
+a P-path from p1 to pk, and write d(p) = p1 and r(p) = pk. We identify each p ∈ P with the
+path (p) of length 1.
+The path category of P is the ∗-category P = P(P) of all P-paths, with object set vP = P,
+under the following operations:
+• For p = (p1, . . . , pk) and q = (q1, . . . , ql) with pk = q1, we define
+p ◦ q = (p1, . . . , pk−1, pk = q1, q2, . . . , ql).
+• For p = (p1, . . . , pk) ∈ P, we define
+prev = (pk, . . . , p1),
+the reverse of p. (It is convenient to write prev instead of p∗, and it is clear that conditions
+(I1)–(I3) from Definition 2.1 all hold.)
+So a morphism set P(p, q) consists of all P-paths from p to q. The identities are the paths
+of the form p ≡ (p). Although P is a ∗-category, it is not a groupoid, as when p has length
+k ≥ 2, p ◦ prev has length 2k − 1 > k. However, and as noted above, a very important role will
+be played by a certain groupoid quotient of P; see Definition 3.39.
+Remark 3.18. It is also worth noting that P is the free ∗-category over the relation F in the
+following sense. We define Γ = ΓP to be the graph with vertex set P, and an edge xpq : p → q for
+each (p, q) ∈ F with p ̸= q. As in [96, p. 49], the free category C = C(Γ) has object set vC = P,
+and its morphisms are the paths in Γ, including the empty path at each vertex p ∈ P (which are
+identified with p, as usual). Non-empty paths can be thought of as words xp1p2xp2p3 · · · xpk−1pk
+(note the matching subscripts between successive letters), and composition of paths is given by
+concatenation when the endpoints match. The involution of C is given by reversal of paths; this
+is well-defined since F is symmetric. It is then clear that
+xp1p2xp2p3 · · · xpk−1pk → (p1, p2, . . . , pk)
+defines a ∗-isomorphism C → P.
+We now wish to show that P is an ordered ∗-category, using Lemma 2.3. To apply the lemma,
+we need a partial order on the object set vP = P, and a collection of (left) restrictions q⇃p.
+We already have the order ≤ on P given in (3.5). To define the restrictions, consider a P-path
+p = (p1, . . . , pk), and suppose q ∈ P is such that q ≤ d(p) = p1. We define a tuple
+q⇃p = (q1, . . . , qk)
+where
+q1 = q
+and
+qi = qi−1θpi for 2 ≤ i ≤ k.
+(3.19)
+35
+
+Note that qi = qθp2 · · · θpi for all 1 ≤ i ≤ k. In fact, since q ≤ p1 we have q = qθp1, so that
+qi = qθp2 · · · θpi = qθp1 · · · θpi
+for all 1 ≤ i ≤ k.
+(3.20)
+It follows immediately that qi ≤ pi for all i.
+Lemma 3.21. If p ∈ P, and if q ≤ d(p), then q⇃p ∈ P.
+Moreover, d(q⇃p) = q and
+r(q⇃p) ≤ r(p).
+Proof. Write p = (p1, . . . , pk) and q⇃p = (q1, . . . , qk), as above. Since p ∈ P, we have pi F pi+1
+for all 1 ≤ i < k, and we must show that qi F qi+1 for all such i, i.e. that
+qiθqi+1 = qi+1
+and
+qi+1θqi = qi.
+For the first, we have
+qiθqi+1 = qiθqiθpi+1 =4 qiθpi+1θqiθpi+1 =3 pi+1θqiθpi+1 =3 qiθpi+1 = qi+1.
+For the second, it follows from (3.20) that qi =1 qiθpi.
+Combining this with pi = pi+1θpi
+(as pi F pi+1), we have
+qi+1θqi = qiθpi+1θqi =3 pi+1θqi = pi+1θqiθpi =4 pi+1θpiθqiθpi = piθqiθpi =3 qiθpi = qi.
+This all shows that q⇃p ∈ P.
+By definition, we have d(q⇃p) = q1 = q, and r(q⇃p) = qk ≤ pk = r(p), where the latter follows
+from the fact, noted above, that qi ≤ pi for all i.
+Lemma 3.22. If p ∈ P and q ≤ d(p), and if r = r(q⇃p), then (q⇃p)rev = r⇃prev.
+Proof. Write p = (p1, . . . , pk) and q⇃p = (q1, . . . , qk), so that r = qk. We also of course have
+prev = (pk, . . . , p1) and (q⇃p)rev = (qk, . . . , q1). To make the subscripts match up, it is convenient
+to write r⇃prev = (rk, . . . , r1). So
+qi = qθp1 · · · θpi
+and
+ri = rθpk · · · θpi
+for all 1 ≤ i ≤ k.
+(3.23)
+We show by descending induction that qi = ri for all i. The i = k case is trivial, as rk = r = qk.
+For 1 ≤ i < k, we have
+ri = ri+1θpi
+by (3.23)
+= qi+1θpi
+by induction
+= qθp1 · · · θpiθpi+1θpi
+by (3.23)
+= qθp1 · · · θpi
+by Lemma 3.12(iv)
+= qi
+by (3.23).
+Lemma 3.24. If p ∈ P and q = d(p), then q⇃p = p.
+Proof. Write p = (p1, . . . , pk), so that q = p1.
+Then q⇃p = (q1, . . . , qk), where each qi =
+qθp1 · · · θpi, and we must show that pi = qi for all i. This is clear for i = 1. For i ≥ 2, we have
+qi = qi−1θpi = pi−1θpi = pi,
+by definition, induction, and the fact that pi F pi−1.
+Lemma 3.25. If p ∈ P and r ≤ q ≤ d(p), then r⇃q⇃p = r⇃p.
+36
+
+Proof. Write
+p = (p1, . . . , pk),
+q⇃p = (q1, . . . , qk),
+r⇃p = (r1, . . . , rk)
+and
+r⇃q⇃p = (s1, . . . , sk).
+So
+qi = qθp1 · · · θpi,
+ri = rθp1 · · · θpi
+and
+si = rθq1 · · · θqi
+for all 1 ≤ i ≤ k,
+(3.26)
+and we must show that ri = si for all i. For i = 1 we have r1 = r = s1. For i ≥ 2,
+si = si−1θqi
+by (3.26)
+= (si−1θqi−1)θqi−1θpi
+by (3.26), noting that si−1 ≤ qi−1
+= si−1θqi−1θpiθqi−1θpi
+by (P4)
+= si−1θqi−1θpi
+by (P5)
+= si−1θpi
+by (3.26)
+= ri−1θpi
+by induction
+= ri
+by (3.26).
+Lemma 3.27. If p, q ∈ P with r(p) = d(q), and if r ≤ d(p) and s = r(r⇃p), then
+r⇃(p ◦ q) = r⇃p ◦ s⇃q.
+Proof. Write p = (p1, . . . , pk) and q = (q1, . . . , ql), noting that pk = q1. Then
+r⇃p = (r, rθp2, rθp2θp3, . . . , rθp2 · · · θpk),
+so
+s = rθp2 · · · θpk.
+It follows that
+r⇃(p ◦ q) = r⇃(p1, . . . , pk, q2, . . . , ql)
+= (r, rθp2, rθp2θp3, . . . , rθp2 · · · θpk = s, sθq2, sθq2θq3, . . . , sθq2 · · · θql)
+= (r, rθp2, rθp2θp3, . . . , rθp2 · · · θpk) ◦ (s, sθq2, sθq2θq3, . . . , sθq2 · · · θql) = r⇃p ◦ s⇃q.
+Proposition 3.28. For any projection algebra P, the path category P = P(P) is an ordered
+∗-category, with ordering given by
+p ≤ q
+⇔
+p = r⇃q
+for some r ≤ d(q).
+Proof. This follows from an application of Lemma 2.3.
+The ordering on vP = P is given
+in (3.5). Properties (O1)′–(O5)′ were established in Lemmas 3.21–3.27.
+As usual (cf. Remark 2.5), we can use the involution to define a right-handed restriction:
+p⇂q = (q⇃prev)rev
+for p ∈ P and q ≤ r(p).
+(3.29)
+Specifically, if p = (p1, . . . , pk) and q ≤ r(p) = pk, then
+p⇂q = (q1, . . . , qk)
+where
+qi = qθpk · · · θpi
+for all 1 ≤ i ≤ k.
+(3.30)
+37
+
+3.3
+Linked pairs
+In the next section we will define the chain groupoid C = C (P) of a projection algebra P as a
+quotient of the path category P = P(P) by a certain congruence ≈. For the definition of ≈ we
+require the concept of linked pairs:
+Definition 3.31. Consider a projection p ∈ P. A pair of projections (e, f) ∈ P 2 is said to be
+p-linked if
+f = eθpθf
+and
+e = fθpθe.
+(3.32)
+Associated to such a p-linked pair (e, f) we define the tuples
+λ(e, p, f) = (e, eθp, f)
+and
+ρ(e, p, f) = (e, fθp, f).
+The next two results gather some important basic properties of p-linked pairs.
+Lemma 3.33. If (e, f) is p-linked, then
+(i) (f, e) is also p-linked, and we have λ(e, p, f)rev = ρ(f, p, e) and ρ(e, p, f)rev = λ(f, p, e),
+(ii) e, f ≤F p,
+(iii) λ(e, p, f) and ρ(e, p, f) both belong to P(e, f).
+Proof. (i). This follows directly by inspecting Definition 3.31.
+(ii). By the symmetry afforded by part (i), it suffices to show that e ≤F p. For this we use (3.32),
+Lemma 3.4 and (P3) to calculate
+pθe = pθfθpθe = pθeθpθfθpθe = eθpθfθpθe = fθpθe = e.
+(iii).
+By symmetry, it suffices to show that λ(e, p, f) ∈ P, i.e. that e F eθp F f.
+From
+e ≤ e ≤F p it follows from Lemma 3.15(ii) that e F eθp. By (3.32) and Lemma 3.12(i), we have
+f = (eθp)θf ≤F eθp, so we are left to show that eθp ≤F f, and for this we use (P4) and (3.32):
+fθeθp = fθpθeθp = eθp.
+Remark 3.34. Consider a projection p ∈ P, and a p-linked pair (e, f). By Lemma 3.33(ii)
+we have e, f ≤F p, and of course we also have eθp, fθp ≤ p. These relationships are all shown
+in Figure 12. In the diagram, each arrow s → t stands for the P-path (s, t) ∈ P. Thus, the
+upper and lower paths from e to f in the bottom part of the diagram correspond to λ(e, p, f)
+and ρ(e, p, f), respectively.
+The next result has an obvious dual, but we will not state it.
+Lemma 3.35. If (e, f) is p-linked, and if e′ ≤ e, then (e′, f′) is p-linked, where f′ = e′θpθf, and
+we have
+e′⇃λ(e, p, f) = λ(e′, p, f′)
+and
+e′⇃ρ(e, p, f) = ρ(e′, p, f′).
+Proof. To show that (e′, f′) is p-linked, we must show that
+f′ = e′θpθf′
+and
+e′ = f′θpθe′.
+For the first we use the definition of f′ and Lemma 3.4 several times to calculate
+e′θpθf′ = e′θpθe′θpθf = e′θpθfθpθe′θpθf = fθpθe′θpθf = e′θpθf = f′.
+38
+
+e
+p
+eθp
+fθp
+f
+Figure 12. A projection p ∈ P, and a p-linked pair (e, f), as in Definition 3.31. Dotted and dashed
+lines indicate ≤F and ≤ relationships, respectively. See Remark 3.34 for more details.
+For the second, we begin with the projection algebra axioms, and calculate
+f′θpθe′ = e′θpθfθpθe′ =4 e′θfθpθe′ =3 fθpθe′ = fθpθeθe′
+by Lemma 3.9, as e′ ≤ e
+= eθe′
+by (3.32)
+= e′
+as e′ ≤F e by Lemma 3.12(ii).
+It remains to show that e′⇃λ = λ′ and e′⇃ρ = ρ′, where for convenience we write
+λ = λ(e, p, f),
+ρ = ρ(e, p, f),
+λ′ = λ(e′, p, f′)
+and
+ρ′ = ρ(e′, p, f′).
+Using (3.19) and Definition 3.31, we have
+e′⇃λ = (e′, e′θeθp, e′θeθpθf),
+λ′ = (e′, e′θp, f′),
+e′⇃ρ = (e′, e′θfθp, e′θfθpθf),
+ρ′ = (e′, f′θp, f′),
+so it remains to check that
+e′θeθp = e′θp,
+e′θeθpθf = f′,
+e′θfθp = f′θp
+and
+e′θfθpθf = f′.
+These are all easily dealt with:
+• e′θeθp =4 e′θpθeθp = e′θeθpθeθp =5 e′θeθp = e′θp, using e′ ≤ e,
+• (e′θeθp)θf = e′θpθf = f′, using the previous calculation,
+• e′θfθp =4 e′θpθfθp = f′θp, and
+• e′θfθpθf =4 e′θpθfθpθf =5 e′θpθf = f′.
+39
+
+3.4
+The chain groupoid
+We are now almost ready to define the chain groupoid C = C (P) associated to a projection
+algebra P. This groupoid is defined below as a certain quotient C = P/≈ of the path category
+P = P(P) from Definition 3.17. The congruence ≈ is defined by specifying a generating set:
+Definition 3.36. Given a projection algebra P (cf. Definition 3.1), let Ω = Ω(P) be the set of
+all pairs (s, t) ∈ P × P of the following three forms:
+(Ω1) s = (p, p) and t = (p) ≡ p, for some p ∈ P,
+(Ω2) s = (p, q, p) and t = (p) ≡ p, for some (p, q) ∈ F,
+(Ω3) s = λ(e, p, f) and t = ρ(e, p, f), for some p ∈ P, and some p-linked pair (e, f).
+We define ≈ = Ω♯ to be the congruence on P generated by Ω.
+Since d(s) = d(t) and r(s) = r(t) for all (s, t) ∈ Ω, it follows that ≈ is a v-congruence.
+Lemma 3.37. The set Ω satisfies (2.18). Consequently, the congruence ≈ satisfies (C4).
+Proof. By Lemma 2.17, it suffices to prove the first claim.
+To do so, we must show that
+(s, t) ∈ Ω
+⇒
+srev ≈ trev for all (s, t) ∈ Ω. This is immediate when (s, t) has the form (Ω1)
+or (Ω2). For (Ω3), it follows from Lemma 3.33(i).
+Lemma 3.38. The set Ω satisfies (2.19). Consequently, the congruence ≈ satisfies (C5).
+Proof. By Lemma 2.17, it suffices to prove the first claim. To do so, let (s, t) ∈ Ω. We must
+show that
+r⇃s ≈ r⇃t
+and
+r(r⇃s) = r(r⇃t)
+for all r ≤ d(s).
+(Note that the equality ϑs = ϑt, which is part of (2.19), is equivalent to r(r⇃s) = r(r⇃t) for all
+r ≤ d(s).) We do this separately for each of the three forms the pair (s, t) can take. This is very
+easy for (Ω1), so we just treat the other two cases.
+(Ω2). Let r ≤ p. To calculate r⇃s = r⇃(p, q, p), we first note that rθqθp = rθpθqθp = rθp = r,
+where we used r ≤ p in the first and third steps, and Lemma 3.12(iv) in the second. We then
+have
+r⇃s = (r, rθq, rθqθp) = (r, rθq, r)
+and of course
+r⇃t = (r).
+This shows that in fact (r⇃s, r⇃t) ∈ Ω, so certainly r⇃s ≈ r⇃t. We also have r(r⇃s) = r = r(r⇃t).
+(Ω3). Let e′ ≤ d(s) = e, and let f′ = e′θpθf. By Lemma 3.35 we have
+e′⇃s = λ(e′, p, f′)
+and
+e′⇃t = ρ(e′, p, f′),
+so again (e′⇃s, e′⇃t) ∈ Ω, and r(e′⇃s) = f′ = r(e′⇃t).
+It follows quickly from iterating (Ω2) that
+p ◦ prev ≈ d(p)
+for all p ∈ P.
+Combining this with Lemmas 2.7, 3.37 and 3.38, it follows that the quotient P/≈ is an ordered
+groupoid.
+Definition 3.39. The chain groupoid of a projection algebra P (cf. Definition 3.1) is the quotient
+C = C (P) = P/≈,
+where P = P(P) is the path category of P (cf. Definition 3.17), and where ≈ is the congruence
+given in Definition 3.36.
+40
+
+• The elements of C , which are ≈-classes of P-paths, are called P-chains. For p ∈ P, we write
+[p] ∈ C for the ≈-class of p. If p = (p1, . . . , pk), then we write [p] = [p1, . . . , pk]. We then have
+d[p] = d(p) = p1 and r[p] = r(p) = pk.
+• For p = (p1, . . . , pk) and q = (q1, . . . , ql) with pk = q1, we have
+[p] ◦ [q] = [p ◦ q] = [p1, . . . , pk−1, pk = q1, q2, . . . , ql].
+• For p = (p1, . . . , pk) ∈ P, we have
+[p]−1 = [prev] = [pk, . . . , p1].
+• The order in C is given by
+c ≤ d
+⇔
+p ≤ q
+for some p ∈ c and q ∈ d.
+Before we move on, it is worth commenting on the definition of the congruence ≈. The chain
+groupoid C = C (P) will be used in Chapter 4 to provide an environment in which to ‘interpret’
+products of projections in certain abstract groupoids G with object set P. Since distinct pro-
+jections/objects have distinct co/domains, they cannot be composed in such a groupoid. But a
+P-chain [p1, . . . , pk] ∈ C is meant to be thought of as a ‘product’ p1 · · · pk, and the ‘interpreta-
+tion’ mentioned above is a functor C → G. Thus, ≈ is meant to equate such ‘products’ when
+they ‘should be equal’ in any regular ∗-semigroup with projection algebra P.
+In this way, it is clear why [p, p] and [p] ‘should’ be equal for any p ∈ P; cf. (Ω1). Similar
+comments apply to [p, q, p] and [p] when p F q; cf. (Ω2) and (2.38). Pairs of the form (Ω3) are
+not quite as obvious. For the purposes of Chapter 4, one could replace ≈ with the congruence ∼
+generated by pairs only of the form (Ω1) and (Ω2), and hence replace C = P/≈ with the
+quotient P/∼. However, as we will see in Chapter 5 (see especially the proof of Lemma 5.15),
+pairs of the form (Ω3) do indeed become equal when interpreted as products of projections in
+any regular ∗-semigroup. At the very least, this means that there is ‘no harm’ in including such
+pairs in the definition of ≈.
+However, there is a more compelling reason for including such pairs. Specifically, we will see
+in Chapter 7 that C = P/≈ gives rise to a ‘free regular ∗-semigroup’ with projection algebra P.
+For this to work, we need C to be a (so-called) chained projection groupoid, and for this we
+require pairs of the form (Ω3) to be equivalent; see especially the proof of Proposition 7.4.
+This of course leads to the question of whether the equivalence of pairs of type (Ω3) is implied
+by those of type (Ω1) and (Ω2), i.e. whether the congruences ≈ and ∼ are in fact equal. But
+it is actually fairly easy to see that they are not. Indeed, it is easy to check that the rewriting
+system generated by the rules
+(p, p) → (p)
+and
+(p, q, p) → (p)
+for p, q ∈ P with p F q
+is complete and Notherian, in the sense of [69]. It follows that any P-path is ∼-equivalent to a
+unique ‘reduced’ path, i.e. one in which there are no sub-paths of the form (p, p) or (p, q, p). This
+reduced path can be obtained by repeatedly but arbitrarily replacing any sub-path of the form
+(p, p) or (p, q, p) by (p). In particular, two P-paths are ∼-equivalent if and only if they have the
+same reduced word. Moreover, it is not hard to show that there are pairs of the form (Ω3) that
+are reduced but not equal, and hence not ∼-equivalent.
+For a concrete example, consider the projection algebra P = P(P3) of the partition monoid P3,
+and define the following projections from P:
+p =
+,
+e =
+and
+f =
+.
+41
+
+One can easily check that (e, f) is p-linked, and that
+eθp = pep =
+and
+fθp = pfp =
+.
+In particular, the paths λ(e, p, f) = (e, eθp, f) and ρ(e, p, f) = (e, fθp, f) are reduced. Since
+eθp ̸= fθp, these reduced paths are not equal, and hence not ∼-equivalent, even though they are
+≈-equivalent by definition; cf. (Ω3).
+Although we will not elaborate on this further, it is worth noting that linked pairs in pro-
+jection algebras are somewhat akin to singular squares in regular biordered sets; cf. [110, p. 20].
+Equating pairs of the form (Ω3) is somewhat akin to requiring singular squares to commute;
+cf. [110, p40].
+3.5
+The category of projection algebras
+Let PA be the (large) category of projection algebras. A morphism in PA is a projection algebra
+morphism φ : P → P ′, by which we mean a map satisfying
+(pθq)φ = (pφ)θ′
+qφ
+for all p, q ∈ P.
+Here we use θ and θ′ to denote the unary operations on P and P ′, respectively. One can also
+think of a projection algebra morphism as a morphism of binary algebras, in the sense discussed
+in Remark 3.2.
+Indeed, using ⋄ and ⋄′ to denote the binary operations of P and P ′ (as in
+Remark 3.2), φ : P → P ′ is a projection algebra morphism if and only if
+(p ⋄ q)φ = (pφ) ⋄′ (qφ)
+for all p, q ∈ P.
+In the next result we also write ≤ and ≤′ for the partial orders on P and P ′, as in (3.5), and
+similarly for the ≤F and F relations from (3.10) and (3.11).
+Lemma 3.40. If φ : P → P ′ is a projection algebra morphism, then for any p, q ∈ P,
+p ≤ q ⇒ pφ ≤′ qφ,
+p ≤F q ⇒ pφ ≤′
+F qφ
+and
+p F q ⇒ pφ F ′ qφ.
+Proof. The proofs are all essentially the same, so we just prove the first. For this we have
+p ≤ q
+⇒
+p = pθq
+⇒
+pφ = (pθq)φ = (pφ)θ′
+qφ
+⇒
+pφ ≤′ qφ.
+The next result shows that any projection algebra morphism naturally induces a functor
+between the corresponding chain groupoids.
+Proposition 3.41. If φ : P → P ′ is a projection algebra morphism, then there is a well-defined
+ordered groupoid functor
+Φ : C (P) → C (P ′)
+given by
+[p1, . . . , pk]Φ = [p1φ, . . . , pkφ].
+Proof. During the proof we write C = C (P) and C ′ = C (P ′), and similarly for P and P′. By
+Lemma 3.40, we have a well-defined functor
+ϕ : P → C ′
+given by
+(p1, . . . , pk)ϕ = [p1φ, . . . , pkφ].
+We first show that ≈ ⊆ ker(ϕ). To do so, it suffices to show that sϕ = tϕ for any pair (s, t) ∈ Ω.
+This is clear if (s, t) has type (Ω1). We now consider the other two types.
+42
+
+(Ω2). Next suppose s = (p, q, p) and t = (p) for some (p, q) ∈ F. By Lemma 3.40, we have
+pφ F qφ (in P ′), and so sϕ = [pφ, qφ, pφ] = [pφ] = tϕ.
+(Ω3).
+Finally, suppose s = λ(e, p, f) = (e, eθp, f) and t = ρ(e, p, f) = (e, fθp, f) for some
+p ∈ P, and some p-linked pair (e, f).
+It is then easy to see that (eφ, fφ) is pφ-linked, and
+sϕ = [λ(eφ, pφ, fφ)] and tϕ = [ρ(eφ, pφ, fφ)], so again we have sϕ = tϕ.
+Now that we know ≈ ⊆ ker ϕ, we have an induced functor
+Φ : C = P/≈ → C ′
+given by
+[p]Φ = pϕ,
+and this is of course the map in the statement of the proposition. It is easy to check that φ being
+a morphism implies (q⇃c)Φ = qφ⇃(cΦ) for all c ∈ C and q ≤ d(c), so that Φ is ordered.
+4
+Projection groupoids
+Recall from Section 2.3 (see Definition 2.32 and Proposition 2.33) that to any regular ∗-semigroup S
+we can associate a groupoid G = G(S). The object set of this groupoid is vG = P = P(S), the
+projection algebra of S, and the (partial) composition in G is a restriction of the (total) product
+in S. Nevertheless, as explained in Remark 2.34, the groupoid G contains enough information
+to completely recover the entire product, in the sense that an arbitrary product ab in S can be
+written as a composition in G, viz. ab = a′ ◦ e ◦ b′ for suitable a′, e, b′ ∈ G(= S). This is not
+without subtleties, however, as discussed in Remark 2.36.
+Roughly speaking, the purpose of the current chapter is to provide an abstract framework
+for the above ideas. In Section 4.1 we introduce projection groupoids (see Definition 4.8) as
+certain ordered groupoids G whose object set vG = P is an (abstract) projection algebra, as
+in Definition 3.1. Section 4.2 introduces the concept of an evaluation map, which is a special
+functor ε : C → G, where C = C (P) is the chain groupoid of P, as in Definition 3.39. We then
+introduce the key idea of a chained projection groupoid (G, ε); here G is a projection groupoid,
+and ε : C → G is an evaluation map obeying a certain coherence condition; see Definition 4.22.
+In Section 4.3 we show that an arbitrary chained projection groupoid (G, ε) gives rise to a
+regular ∗-semigroup S = S(G, ε) on the same underlying set as G, with product • extending the
+composition ◦, with involution given by groupoid inversion, and with projection algebra P(S)
+equal to the object set P = vG; see Theorem 4.36. We conclude, in Section 4.4, with some
+basic properties concerning • products of projections in S, including a characterisation of the
+idempotent-generated subsemigroup E(S) = ⟨E(S)⟩ in Proposition 4.39.
+In coming chapters we will take this idea much further.
+Specifically, in Chapter 5 we
+show, conversely, that any regular ∗-semigroup S gives rise to a chained projection groupoid
+G(S) = (G, ε). In Chapter 6 we show that S and G are in fact mutually inverse functors be-
+tween the categories of regular ∗-semigroups and chained projection groupoids (with suitable
+morphisms), so that these categories are isomorphic; see Theorem 6.7.
+4.1
+Projection groupoids
+Let G be an ordered groupoid, and suppose the object set P = vG is a projection algebra (cf. Def-
+inition 3.1), with the ordering on G inherited from that of P (cf. (3.5)) in the sense described in
+Lemma 2.3. That is, for each a ∈ G and each p ≤ d(a) we have a left restriction p⇃a ∈ G, with re-
+spect to which conditions (O1)′–(O5)′ hold. We also have the right restrictions a⇂q = (q⇃a−1)−1,
+defined for q ≤ r(a), and these satisfy the duals of (O1)′–(O5)′. We typically use these conditions
+without explicit reference, and also (O6)′ and its dual, which follow from the others.
+At this point, the only ‘link’ between the structures of the groupoid G and the projection
+algebra P = vG is via the order ≤ on P. This order is defined in terms of the θ operations
+43
+
+in (2.26). Conversely, however, we cannot recover the θ operations from the order ≤; indeed, we
+will see in Example 5.21 that different projection algebras (with the same underlying set) can
+give rise to exactly the same ordering. Thus, we will be particularly interested in groupoids G
+(as above) with a stronger link to their object algebra P = vG. Such a link manifests itself in
+a number of equivalent properties listed in Proposition 4.6 below. But before we get to these
+properties, we first consider the more general situation in which the only assumed link between G
+and P = vG is via the order ≤ on P, as above.
+Consider a morphism a ∈ G. As in (2.10), we have the map
+ϑa : d(a)↓ → r(a)↓
+given by
+pϑa = r(p⇃a).
+Since d(a) ∈ P, we also have the unary operation θd(a), and by (3.8) its image is d(a)↓. It follows
+that we can compose θd(a) with ϑa, and we denote this composition by
+Θa = θd(a)ϑa : P → r(a)↓.
+(4.1)
+As a special case, note that by Lemma 2.14(ii), we have
+Θp = θd(p)ϑp = θp idp↓ = θp
+for any projection p ∈ P.
+(4.2)
+We begin by recording some basic properties of the Θ maps.
+Lemma 4.3. For any a ∈ G we have:
+(i) ϑaθr(a) = ϑa,
+(ii) Θaθr(a) = Θa = θd(a)Θa,
+(iii) Θaϑa−1 = θd(a),
+(iv) Θa = θd(a)ϑb for any a ≤ b,
+(v) Θp⇃a = θpΘa for any p ≤ d(a).
+Proof. (i). Since im(ϑa) = r(a)↓, this follows immediately from the fact that each operation θp
+fixes each element of im(θp) = p↓.
+(ii). Since Θa = θd(a)ϑa, we obtain Θaθr(a) = Θa from part (i), and θd(a)Θa = Θa from (P2).
+(iii). Since im(θd(a)) = d(a)↓, it follows from Lemma 2.12 that
+Θaϑa−1 = θd(a)ϑaϑa−1 = θd(a) idd(a)↓ = θd(a).
+(iv). Since a ≤ b, we have a = p⇃b, where p = d(a). Now let s ��� P be arbitrary; we must show
+that sΘa = sθpϑb. For this, we write t = sθp, and use (O4)′ to calculate
+sΘa = sθpϑa = tϑa = r(t⇃a) = r(t⇃p⇃b) = r(t⇃b) = tϑb = sθpϑb.
+(v). We have
+Θp⇃a = θd(p⇃a)ϑa = θpϑa
+by part (iv), as p⇃a ≤ a
+= θpθd(a)ϑa
+by Lemma 3.9, as p ≤ d(a)
+= θpΘa.
+Another important property is that when a and b are composable in G, the map Θa◦b is the
+composition of Θa and Θb.
+44
+
+Lemma 4.4. If a, b ∈ G are such that r(a) = d(b), then Θa◦b = ΘaΘb.
+Proof. Write p = d(a) and q = r(a) = d(b). Then by Lemmas 2.14(i) and 4.3(i) we have
+ΘaΘb = θpϑaθqϑb = θpϑaθr(a)ϑb = θpϑaϑb = θpϑa◦b = Θa◦b.
+Part (v) of Lemma 4.3 above shows how the Θ maps interact with left restrictions, but the
+lemma did not include a corresponding statement concerning right restrictions.
+Of course if
+q ≤ r(a), then a⇂q = p⇃a where p = qϑa−1 (cf. (2.11)), and then
+Θa⇂q = Θp⇃a = θpΘa = θqϑa−1Θa.
+(4.5)
+However, the groupoids we will be concerned with satisfy the neater identity Θa⇂q = Θaθq. This
+is in fact equivalent to a number of additional properties linking the groupoid structure of G to
+the projection algebra structure of P = vG:
+Proposition 4.6. If G is an ordered groupoid, whose object set P = vG is a projection algebra,
+as above, then the following are equivalent:
+(G1a) θpϑa = Θa−1θpΘa for all a ∈ G and p ≤ d(a),
+(G1b) θpΘa = Θa−1θpΘa for all a ∈ G and p ∈ P,
+(G1c) Θa⇂q = Θaθq for all a ∈ G and q ≤ r(a),
+(G1d) ϑa is a projection algebra morphism (and hence isomorphism) d(a)↓ → r(a)↓ for all a ∈ G.
+Proof. For the duration of the proof we fix a ∈ G, and we write s = d(a) and t = r(a). On a
+number of occasions we will use the fact that
+Θa−1Θa = Θa−1◦a = Θr(a) = Θt = θt.
+(4.7)
+In the above calculation we used Lemma 4.4 and (4.2). In what follows, we show that (G1a)
+implies each of (G1b)–(G1d), and conversely that any of (G1b)–(G1d) implies (G1a).
+(G1a) ⇒ (G1b)–(G1d). Suppose first that (G1a) holds. To verify (G1b), let p ∈ P be arbitrary.
+Then
+θpΘa = θ(pθs)ϑa
+by definition
+= Θa−1θpθsΘa
+by (G1a), as pθs ≤ s = d(a)
+= Θa−1θsθpθsΘa
+by (P4)
+= Θa−1θr(a−1)θpθd(a)Θa
+= Θa−1θpΘa
+by Lemma 4.3(ii).
+For (G1c), if q ≤ r(a) then
+Θa⇂q = θqϑa−1Θa
+by (4.5)
+= ΘaθqΘa−1Θa
+by (G1a)
+= Θaθqθr(a)
+by (4.7)
+= Θaθq
+by Lemma 3.9, as q ≤ r(a).
+For (G1d), since ϑa is a bijection s↓ → t↓ by Lemma 2.12, it is enough to show that ϑa is a
+projection algebra morphism, i.e. that
+(pθq)ϑa = (pϑa)θqϑa
+for all p, q ≤ s = d(a).
+45
+
+But for any such p, q, we have
+(pϑa)θqϑa = pϑaΘa−1θqΘa
+by (G1a)
+= pϑaθtϑa−1θqθsϑa
+by definition of the Θ maps
+= pϑaϑa−1θqϑa
+as pϑa ≤ r(a) = t and q ≤ s (cf. Lemma 3.9)
+= pθqϑa
+by Lemma 2.12.
+(G1b) ⇒ (G1a). This is immediate, since for any p ≤ d(a) = s we have pϑa = (pθs)ϑa = pΘa .
+(G1c) ⇒ (G1a). If (G1c) holds, then for any p ≤ d(a) = r(a−1) we have
+Θa−1θpΘa = Θa−1⇂pΘa
+by (G1c)
+= θpϑaΘa−1Θa
+by (4.5)
+= θpϑaθr(a)
+by (4.7)
+= θpϑa
+by Lemma 3.9, as pϑa ≤ r(a).
+(G1d) ⇒ (G1a). Suppose (G1d) holds, and let p ≤ s. We must show that
+eθpϑa = eΘa−1θpΘa
+for any e ∈ P,
+so fix some such e.
+To make the following calculation easier to read, let f = eθt.
+Since
+f ≤ t = d(a−1), we can also define g = fϑa−1.
+By Lemma 2.12 we have f = gϑa, and we
+note also that g ≤ r(a−1) = s. We then have
+eθpϑa = eθtθpϑa = fθpϑa
+by Lemma 3.9, as pϑa ≤ r(a) = t
+= (gϑa)θpϑa
+as observed above
+= (gθp)ϑa
+by (G1d), as g, p ≤ s = d(a)
+= eθtϑa−1θpϑa
+as g = fϑa−1 and f = eθt
+= eθtϑa−1θpθsϑa
+by Lemma 3.9, as p ≤ s
+= eΘa−1θpΘa
+by definition of the Θ maps.
+Here then is the main definition of this section:
+Definition 4.8. A projection groupoid is an ordered groupoid G, whose object set P = vG is a
+projection algebra (cf. Definition 3.1), with the ordering on G inherited from that of P (cf. (3.5))
+in the sense described in Lemma 2.3, and for which:
+(G1) any (and hence all) of the conditions (G1a)–(G1d) hold.
+Remark 4.9. The reader should notice a resemblance between (G1a) and (P4). In fact, this is
+more than just superficial. Recall that (abstract) projection algebras are supposed to ‘model’ pro-
+jections of regular ∗-semigroups, and that the θp operations model ‘conjugation’, viz. qθp ≡ pqp.
+In this way, axiom (P4) can be thought of as a recipe for ‘iterating’ conjugation by projections.
+By the same token, restrictions p⇃a in G (with p ≤ d(a)) are supposed to model ‘restric-
+tions’ pa in a regular ∗-semigroup S, as in Remark 2.34. ‘Pretending’ that we are working in a
+regular ∗-semigroup, we can then think of
+pϑa = r(p⇃a) ≡ r(pa) ≡ (pa)∗pa = a∗p∗pa = a∗pa
+46
+
+as a kind of ‘conjugate of p by a’. In this sense, the ϑa maps can be thought of as analogues of
+conjugation in an arbitrary (abstract) projection groupoid. For arbitrary t ∈ P (not necessarily
+below d(a)), we can then also think of
+tΘa = tθd(a)ϑa ≡ a∗ · d(a) · t · d(a) · a ≡ a∗ · aa∗ · t · aa∗ · a = a∗ta,
+so that Θa also acts as a kind of ‘conjugation’ by morphisms. Condition (G1a) can then be
+thought of as a way to iterate this kind of conjugation, as can condition (G1b).
+We close the section with a simple technical result that will help us shorten some later
+arguments.
+Lemma 4.10. If G is a projection groupoid, and if a ∈ G and p, q ∈ P, then
+pΘa−1θqΘaθp = qΘaθp.
+Proof. By (G1b) and (P3) we have pΘa−1θqΘaθp = pθqΘaθp = qΘaθp.
+4.2
+Chained projection groupoids
+At this point it is worth pausing to consider again the broad goal of this chapter, and to draw
+attention to one of the major hurdles we currently face. Recall that (broadly speaking) we are
+aiming to give an abstract description of the groupoids that correspond to regular ∗-semigroups.
+Starting with a regular ∗-semigroup S, in Definition 2.32 we constructed a groupoid G = G(S)
+with underlying set S, and whose composition was a restriction of the product in S. As we
+explained in Remark 2.34, this (partial) composition is enough to reconstruct the entire product
+of S, in the sense that for any a, b ∈ S we have ab = a′ ◦ e ◦ b′ for suitable a′, e, b′ ∈ G(= S).
+As noted in that remark, the element e is in fact the idempotent e = p′q′, where p′, q′ ∈ P are
+certain special projections below r(a) = a∗a and d(b) = bb∗. Being an idempotent of course
+means that e2 = e in S. However, since d(e) = p′ and r(e) = q′ (cf. (2.41)), and since p′ and q′
+are not necessarily equal (though always F-related), the composition e◦e is generally not defined
+in G. Returning to the abstract setting of projection groupoids, in which we are operating in
+this chapter, it is not at all obvious at the outset which morphism p′ → q′ ought to play the role
+of this idempotent e. The precise mechanism for solving this problem (which was explored at
+greater length in Remark 2.36) is encoded in what we will call an evaluation map, and in the
+properties we will require of them below.
+Definition 4.11. Let G be a projection groupoid (cf. Definition 4.8), and let C = C (P) be
+the chain groupoid of P = vG (cf. Definition 3.39). An evaluation map is an ordered v-functor
+ε : C → G, meaning that the following hold:
+(E1) ε(p) = p for all p ∈ P (where as usual we identify p ≡ [p] ∈ C ),
+(E2) ε(c ◦ d) = ε(c) ◦ ε(d) if r(c) = d(d),
+(E3) c ≤ d ⇒ ε(c) ≤ ε(d).
+We will soon define a chained projection groupoid to be a projection groupoid with an eval-
+uation map possessing a certain coherence property.
+But first we note a number of simple
+consequences of (E1)–(E3).
+Remark 4.12. As with group homomorphisms, it is easy to see that
+(E4) ε(c−1) = ε(c)−1 for all c ∈ C .
+47
+
+It also follows that:
+(E5) d(ε(c)) = d(c) and r(ε(c)) = r(c) for all c ∈ C .
+For example, d(ε(c)) = ε(c) ◦ ε(c)−1 = ε(c) ◦ ε(c−1) = ε(c ◦ c−1) = ε(d(c)).
+It is also easy to see that (E3) is equivalent (in the presence of the other axioms) to:
+(E6) ε(q⇃c) = q⇃ε(c) if q ≤ d(c).
+For example, if (E6) holds, and if c ≤ d, then c = p⇃d for some p ≤ d(d); it follows from this that
+ε(c) = ε(p⇃d) = p⇃ε(d) ≤ ε(d).
+Consider a projection groupoid G with P = vG, and let ε : C → G be an evaluation map.
+For p, q ∈ P with p F q, we have the P-chain [p, q] ∈ C , and the elements ε[p, q] ∈ G will play
+a crucial role in all that follows. For one thing, these elements generate the image of C under ε
+(as the [p, q] generate C ). Specifically, if c = [p1, p2, . . . , pk] ∈ C , then
+ε(c) = ε[p1, p2] ◦ ε[p2, p3] ◦ · · · ◦ ε[pk−1, pk].
+(4.13)
+The next lemma gathers some important basic properties of the ε[p, q]; in what follows, we will
+typically use these without explicit reference. We write f|A for the (set-theoretic) restriction of
+a function f to a subset A of its domain.
+Lemma 4.14. Let G be a projection groupoid, and ε : C → G an evaluation map.
+(i) For any p ∈ P we have ε[p, p] = p.
+(ii) If p, q ∈ P are such that p F q, then
+d(ε[p, q]) = p,
+ε[p, q]−1 = ε[q, p],
+ϑε[p,q] = θq|p↓,
+r(ε[p, q]) = q,
+Θε[p,q] = θpθq.
+(iii) If p, q, r, s ∈ P are such that p F q, r ≤ p and s ≤ q, then
+r⇃ε[p, q] = ε[r, rθq]
+and
+ε[p, q]⇂s = ε[sθp, s].
+Proof. (i). This follows quickly from (E1) and [p, p] = [p], keeping in mind the identification of
+p ∈ P with the chain [p] ∈ C .
+(iii). This follows from (E6), together with (3.19) and (3.30).
+(ii). The claims concerning domains, ranges and inversion follow from (E4) and (E5), and the
+fact that [p, q]−1 = [q, p] in C .
+Since the maps ϑε[p,q] and θq|p↓ both have domain p↓, we can show the maps are equal by
+showing that
+rϑε[p,q] = rθq
+for all r ≤ p.
+But for such r, we use the definition of the ϑ maps, and parts of the current lemma that have
+already been proved, to calculate
+rϑε[p,q] = r(r⇃ε[p, q]) = r(ε[r, rθq]) = rθq.
+Finally, again using already-proved parts of the current lemma, we have
+Θε[p,q] = θpϑε[p,q] = θp ◦ θq|p↓ = θpθq,
+where in the last step we used the fact that p↓ = im(θp).
+48
+
+Before we move on, it will also be convenient to record the following result.
+Lemma 4.15. Let G be a projection groupoid, and ε : C → G an evaluation map. If p, q ∈ P
+are such that p F q, and if a ≤ ε[p, q], then
+a = ε[r, s]
+where
+r = d(a)
+and
+s = r(a).
+Proof. Noting that a ≤ ε[p, q], we use Lemma 4.14 to calculate
+a = r⇃ε[p, q] = ε[r, rθq]
+and
+rθq = r(ε[r, rθq]) = r(a) = s.
+To state the coherence property alluded to above, we require the concept of linked pairs,
+which we now define. We have reused this term from Section 3.3 (where p-linked pairs were used
+to define the congruence ≈ on the path category P), because there is a strong tie between the
+two concepts, as we will see later.
+Definition 4.16. Let G be a projection groupoid, and consider a morphism b ∈ G. A pair of
+projections (e, f) ∈ P 2 is said to be b-linked if
+f = eΘbθf
+and
+e = fΘb−1θe.
+(4.17)
+We will soon associate two morphisms λ(e, b, f) and ρ(e, b, f) to the b-linked pair (e, f). But
+first we prove the next result, which will ensure these morphisms are well defined.
+Lemma 4.18. Let G be a projection groupoid, and suppose (e, f) is b-linked, where b ∈ G(q, r).
+Define further projections
+e1 = eθq,
+e2 = fΘb−1,
+f1 = eΘb
+and
+f2 = fθr.
+(4.19)
+Then
+(i) e ≤F q and f ≤F r,
+(ii) ei ≤ q and fi ≤ r for i = 1, 2,
+(iii) e F ei and f F fi for i = 1, 2,
+(iv) ei⇃b = b⇂fi for i = 1, 2.
+Proof. It is clear from Definition 4.16 that (e, f) is b-linked if and only if (f, e) is b−1-linked. If
+we replace b ↔ b−1 and (e, f) ↔ (f, e), then the projections defined in (4.19) are interchanged
+accordingly, viz. e1 ↔ f2 and e2 ↔ f1. Because of this symmetry, for parts (i)–(iii), it suffices to
+prove the claims concerning e, e1, e2. (Alternatively, the arguments below can be easily adapted
+to prove the claims regarding f, f1, f2.)
+(ii). This follows from im(Θb−1) = im(θq) = q↓.
+(iii). From e ≤ e ≤F q, it follows from Lemma 3.15(ii) that e F eθq = e1. By Lemma 3.12(i),
+we have e = e2θe ≤F e2. It remains to show that e2 ≤F e, and for this we use (G1b), (4.17)
+and (4.19):
+eθe2 = eθfΘb−1 = eΘbθfΘb−1 = fΘb−1 = e2.
+(iv). We must show that fi = eiϑb (i = 1, 2). Keeping q = d(b) and r = d(b−1) in mind, we have
+e1ϑb = eθqϑb = eΘb = f1
+and
+e2ϑb = fΘb−1ϑb = fθr = f2,
+where we used Lemma 4.3(iii) in the second calculation.
+(i). Combining parts (ii) and (iii), we have e ≤F e1 ≤ q, and Lemma 3.15(i) then gives e ≤F q.
+49
+
+Definition 4.20. Let G be a projection groupoid (cf. Definition 4.8), and ε : C → G an evaluation
+map (cf. Definition 4.11). Let (e, f) be a b-linked pair, where b ∈ G(q, r), and let e1, e2, f1, f2 ∈ P
+be as in (4.19). By Lemma 4.18, we have two well-defined morphisms
+λ(e, b, f) = ε[e, e1] ◦ e1⇃b ◦ ε[f1, f]
+and
+ρ(e, b, f) = ε[e, e2] ◦ e2⇃b ◦ ε[f2, f]
+= ε[e, e1] ◦ b⇂f1 ◦ ε[f1, f]
+= ε[e, e2] ◦ b⇂f2 ◦ ε[f2, f].
+(4.21)
+These morphisms are shown in Figure 13.
+e
+e1
+e2
+f
+f1
+f2
+q
+r
+b
+e1⇃b = b⇂f1
+e2⇃b = b⇂f2
+ε[e, e1]
+ε[e, e2]
+ε[f1, f]
+ε[f2, f]
+Figure 13. The projections and morphisms associated to a b-linked pair (e, f), for b ∈ G(q, r); see
+Definitions 4.16, 4.20 and 4.22, and Lemma 4.18. Dotted and dashed lines indicate ≤F and ≤ rela-
+tionships, respectively. Axiom (G2) says that the hexagon at the bottom of the diagram commutes.
+The above-mentioned coherence property states that the two terms in (4.21) must be equal:
+Definition 4.22. A chained projection groupoid is a pair (G, ε), where G is a projection groupoid
+(cf. Definition 4.8), and ε : C → G is an evaluation map (cf. Definitions 3.39 and 4.11) satisfying
+the following condition:
+(G2) For every b ∈ G, and for every b-linked pair (e, f), we have λ(e, b, f) = ρ(e, b, f), where
+these morphisms are as in (4.21).
+Remark 4.23. One might wonder if the concept of linked pairs could be avoided when defining
+chained projection groupoids. Specifically, one might wonder if (G2) is equivalent to:
+(G2)′ For any morphism b ∈ G(q, r), and for any projections e, e1, e2, f, f1, f2 ∈ P satisfying
+conditions (i)–(iv) from Lemma 4.18, we have ε[e, e1]◦e1⇃b◦ε[f1, f] = ε[e, e2]◦e2⇃b◦ε[f2, f].
+While (G2)′ of course implies (G2), the converse does not hold. Most significantly, the stronger
+condition (G2)′ does not (generally) hold in the all-important case that G = G(S) is the groupoid
+associated to a regular ∗-semigroup S, as in Definition 2.32. We will say more about this in
+Remark 5.20.
+50
+
+4.3
+The regular ∗-semigroup associated to a chained projection groupoid
+For the duration of this section, we fix a chained projection groupoid (G, ε), as in Definition 4.22,
+and we continue to write P = vG, C = C (P), and so on. Our aim here is to construct a regular
+∗-semigroup S = S(G, ε), built from G and ε in a natural way. The underlying set of S will
+simply be G, and the involution of S will simply be inversion in G:
+a∗ = a−1
+for a ∈ G.
+The definition of the product in S, which we will denote by •, is more involved. Its inspiration
+is drawn from the properties of regular ∗-semigroups discussed in Remark 2.34.
+Definition 4.24. Let (G, ε) be a chained projection groupoid (cf. Definition 4.22), and consider
+an arbitrary pair of morphisms a, b ∈ G. Let p = r(a) and q = d(b), and define the projections
+p′ = qθp
+and
+q′ = pθq.
+By Lemma 3.14, we have p′ ≤ p, q′ ≤ q and p′ F q′. In particular, the morphisms a⇂p′, ε[p′, q′]
+and q′⇃b exist, and we define
+a • b = a⇂p′ ◦ ε[p′, q′] ◦ q′⇃b.
+(4.25)
+This is illustrated in Figure 14. We define S(G, ε) to be the (2, 1)-algebra with underlying set G,
+and with binary operation • and unary operation ∗ = −1.
+p
+q
+p′
+q′
+a
+b
+a⇂p′
+q′⇃b
+ε[p′, q′]
+a • b
+Figure 14. Construction of the product a • b, as in Definition 4.24.
+To show that S = S(G, ε) is a regular ∗-semigroup, with respect to the binary operation •
+and unary operation ∗ = −1, we must establish the identities:
+(a • b) • c = a • (b • c),
+(a∗)∗ = a = (a • a∗) • a
+and
+(a • b)∗ = b∗ • a∗.
+Associativity of • is quite difficult to establish, and is finally achieved in Lemma 4.34 below. The
+identities involving ∗ are comparatively easy, however, and we verify these shortly in Lemma 4.27.
+For its proof, we need the following lemma, which shows that the (total) product • extends the
+(partial) composition ◦, in the sense that these operations agree whenever the latter is defined.
+Before we begin, it is worth noting that the operation • from Definition 4.24 makes sense in
+the more general case of G being a projection groupoid with an evaluation map ε. All of the
+coming lemmas apart from the final Lemma 4.34 hold in this more general situation as well; the
+coherence property (G2) is needed only at the very last stage of the proof of associativity. Still,
+for simplicity, we do continue to assume that (G, ε) is a chained projection groupoid.
+51
+
+Lemma 4.26. If a, b ∈ G are such that r(a) = d(b), then a • b = a ◦ b.
+Proof. In the notation of Definition 4.24, we have p = q, so also p′ = q′ = p. But then
+a • b = a⇂p ◦ ε[p, p] ◦ p⇃b = a ◦ p ◦ b = a ◦ b.
+Lemma 4.27. For any a, b ∈ G, we have
+(a∗)∗ = a = (a • a∗) • a
+and
+(a • b)∗ = b∗ • a∗.
+Proof. Of course (a∗)∗ = (a−1)−1 = a. It follows from Lemma 4.26 that a•a∗ = a◦a−1 = d(a),
+and so (a • a∗) • a = d(a) • a = d(a) ◦ a = a.
+Next, let p, q, p′, q′ be as in Definition 4.24. Since q = r(b∗) and p = d(a∗), we have
+a • b = a⇂p′ ◦ ε[p′, q′] ◦ q′⇃b
+and
+b∗ • a∗ = b∗⇂q′ ◦ ε[q′, p′] ◦ p′⇃a∗,
+and so
+(a • b)∗ = (q′⇃b)∗ ◦ ε[p′, q′]∗ ◦ (a⇂p′)∗ = b∗⇂q′ ◦ ε[q′, p′] ◦ p′⇃a∗ = b∗ • a∗.
+The symmetry/duality afforded by the identity (a • b)∗ = b∗ • a∗ will allow for some simplifi-
+cation in some of the proofs to follow. This is the case with the proof of the next result, which
+identifies the domain and range of a product a • b.
+Lemma 4.28. For any a, b ∈ G we have
+d(a • b) = d(b)Θa−1
+and
+r(a • b) = r(a)Θb.
+Proof. Let p, q, p′, q′ be as in Definition 4.24, so that a • b is as in (4.25). Then
+r(a • b) = r(q′⇃b) = q′ϑb = pθqϑb = r(a)θd(b)ϑb = r(a)Θb.
+The identity involving domains may be proved in similar fashion (using (2.11)). Alternatively,
+it follows from the range identity and duality:
+d(a • b) = r((a • b)∗) = r(b∗ • a∗) = r(b∗)Θa∗ = d(b)Θa−1.
+Our next result establishes an important identity.
+Lemma 4.29. For any a, b ∈ G we have Θa•b = ΘaΘb.
+Proof. Let p, q, p′, q′ be as in Definition 4.24. Then by the projection algebra axioms we have
+θp′θq′ = θqθpθpθq =4 θpθqθpθqθpθq =5 θpθq.
+We then calculate
+Θa•b = Θa⇂p′ ◦ Θε[p′,q′] ◦ Θq′⇃b
+by (4.25) and Lemma 4.4
+= Θaθp′ ◦ θp′θq′ ◦ θq′Θb
+by (G1c) and Lemmas 4.3(v) and 4.14(ii)
+= Θa ◦ θp′θq′ ◦ Θb
+by (P2)
+= Θa ◦ θpθq ◦ Θb
+by the above observation
+= Θaθr(a) ◦ θd(b)Θb
+= ΘaΘb
+by Lemma 4.3(ii).
+52
+
+We are almost ready to show that • is associative, i.e. that (a • b) • c and a • (b • c) are equal,
+for all a, b, c ∈ G. Before we do so, we note that Lemmas 4.28 and 4.29 can be used to show that
+these two terms have equal domains, and equal ranges. For example,
+r((a • b) • c) = r(a • b)Θc = r(a)ΘbΘc = r(a)Θb•c = r(a • (b • c)).
+(4.30)
+A similar calculation gives
+d((a • b) • c) = d(a �� (b • c)) = d(c)Θb−1Θa−1.
+(4.31)
+As a stepping stone to associativity, the next result gives expansions of (a•b)•c and a•(b•c),
+expressing each term as a composition of five morphisms.
+Lemma 4.32. If a, b, c ∈ G, and if p = r(a), q = d(b), r = r(b) and s = d(c), then
+(i) (a • b) • c = a⇂e ◦ ε[e, e1] ◦ e1⇃b ◦ ε[f1, f] ◦ f⇃c, where
+e = sΘb−1θp,
+e1 = eθq,
+f1 = eΘb
+and
+f = pΘbθs,
+(ii) a • (b • c) = a⇂e ◦ ε[e, e2] ◦ b⇂f2 ◦ ε[f2, f] ◦ f⇃c, where
+e = sΘb−1θp,
+e2 = fΘb−1,
+f2 = fθr
+and
+f = pΘbθs.
+The above factorisations are illustrated in Figure 15.
+Proof. (i). Put p′ = qθp and q′ = pθq, so that
+a • b = a′ ◦ ε′ ◦ b′
+where
+a′ = a⇂p′,
+ε′ = ε[p′, q′]
+and
+b′ = q′⇃b.
+Keeping Lemma 4.28 in mind, we now define
+t = r(a • b) = pΘb.
+Then
+(a • b) • c = (a • b)⇂t′ ◦ ε[t′, s′] ◦ s′⇃c
+where
+t′ = sθt and s′ = tθs.
+We now focus on the term (a • b)⇂t′. By (2.13) we have
+(a • b)⇂t′ = (a′ ◦ ε′ ◦ b′)⇂t′ = a′⇂u ◦ ε′⇂v ◦ b′⇂t′
+where
+v = t′ϑ−1
+b′
+and
+u = t′ϑ−1
+b′ ϑ−1
+ε′ .
+Now,
+• a′⇂u = a⇂p′⇂u = a⇂u,
+• ε′⇂v ≤ ε′ = ε[p′, q′], d(ε′⇂v) = r(a′⇂u) = u and r(ε′⇂v) = v, so ε′⇂v = ε[u, v] by Lemma 4.15,
+and
+• b′⇂t′ ≤ b′ ≤ b and d(b′⇂t′) = r(ε′⇂v) = v, so b′⇂t′ = v⇃b. (For later use, since b′⇂t′ and b⇂t′ are
+both below b and have codomain t′, it also follows that b⇂t′ = b′⇂t′ = v⇃b.)
+Putting everything together, it follows that
+(a • b) • c = (a • b)⇂t′ ◦ ε[t′, s′] ◦ s′⇃c
+= a′⇂u ◦ ε′⇂v ◦ b′⇂t′ ◦ ε[t′, s′] ◦ s′⇃c
+= a⇂u ◦ ε[u, v] ◦ v⇃b ◦ ε[t′, s′] ◦ s′⇃c.
+53
+
+It therefore remains to check that
+(a) u = e,
+(b) v = e1,
+(c) t′ = f1,
+(d) s′ = f,
+where e, e1, f1, f are as defined in the statement of the lemma.
+(a). First note that d(a⇂u) = d((a • b) • c) = sΘb−1Θa−1 by (4.31), so a⇂u = sΘb−1Θa−1⇃a. But
+then
+u = r(a⇂u) = r(sΘb−1Θa−1⇃a) = sΘb−1Θa−1ϑa = sΘb−1θp = e,
+where we used Lemma 4.3(iii) in the second-last step.
+(c). Here we use (G1b) to calculate t′ = sθt = sθpΘb = sΘb−1θpΘb = eΘb = f1.
+(b). We noted above that v⇃b = b⇂t′. Combining this with (2.11), the just-proved item (c), and
+Lemma 4.3(iii), it follows that
+v = d(v⇃b) = d(b⇂t′) = t′ϑb−1 = f1ϑb−1 = (eΘb)ϑb−1 = eθq = e1.
+(d). Finally, s′ = tθs = pΘbθs = f.
+As noted above, this completes the proof of (i).
+(ii). This can be proved in similar fashion to part (i), but in fact we can also deduce it from (i).
+Indeed, we first use Lemma 4.27 to expand
+(a • (b • c))−1 = (b • c)−1 • a−1 = (c−1 • b−1) • a−1.
+Note that s = r(c−1), r = d(b−1), q = r(b−1) and p = d(a−1). It follows from part (i) that
+(c−1 • b−1) • a−1 = c−1⇂e′ ◦ ε[e′, e′
+1] ◦ e′
+1⇃b−1 ◦ ε[f′
+1, f′] ◦ f′⇃a−1,
+where
+e′ = pΘbθs,
+e′
+1 = e′θr,
+f′
+1 = e′Θb−1
+and
+f′ = sΘb−1θp.
+But then
+a • (b • c) = ((c−1 • b−1) • a−1)−1
+= (c−1⇂e′ ◦ ε[e′, e′
+1] ◦ e′
+1⇃b−1 ◦ ε[f′
+1, f′] ◦ f′⇃a−1)−1
+= (f′⇃a−1)−1 ◦ ε[f′
+1, f′]−1 ◦ (e′
+1⇃b−1)−1 ◦ ε[e′, e′
+1]−1 ◦ (c−1⇂e′)−1
+= a⇂f′ ◦ ε[f′, f′
+1] ◦ b⇂e′
+1 ◦ ε[e′
+1, e′] ◦ e′⇃c.
+(4.33)
+We note immediately that f′ = e and e′ = f (where e, f are as in the statement of the lemma).
+We also have
+e′
+1 = e′θr = fθr = f2
+and
+f′
+1 = e′Θb−1 = fΘb−1 = e2.
+Thus, continuing from (4.33), we have
+a • (b • c) = a⇂e ◦ ε[e, e2] ◦ b⇂f2 ◦ ε[f2, f] ◦ f⇃c,
+as required.
+54
+
+e
+e1
+e2
+f
+f1
+f2
+p
+q
+r
+s
+a
+b
+c
+e1⇃b = b⇂f1
+e2⇃b = b⇂f2
+ε[e, e1]
+ε[e, e2]
+ε[f1, f]
+ε[f2, f]
+a⇂e
+f⇃c
+Figure 15. Top: three morphisms a, b, c ∈ G. Bottom: the upper and lower paths represent (a•b)•c
+and a • (b • c), respectively; cf. Lemma 4.32. Dashed lines indicate ≤ relationships.
+Lemma 4.34. For any a, b, c ∈ G, we have (a • b) • c = a • (b • c).
+Proof. By Lemma 4.32, it suffices to show (in the notation of that lemma) that
+ε[e, e1] ◦ e1⇃b ◦ ε[f1, f] = ε[e, e2] ◦ b⇂f2 ◦ ε[f2, f].
+(4.35)
+Comparing the definitions of the ei, fi from Lemma 4.32 (in terms of e, f, q, r, b) with (4.19), an
+application of (G2) will give us (4.35) if we can show that (e, f) is b-linked (cf. Definitions 4.16
+and 4.20). In other words, the proof of the lemma will be complete if we can show that
+f = eΘbθf
+and
+e = fΘb−1θe.
+For the first, we have
+eΘbθf = sΘb−1θpΘbθpΘbθs
+by definition
+= sΘb−1θpΘbθsθpΘbθs
+by (P4)
+= pΘbθsΘb−1θpΘbθs
+by (G1b) and Lemma 4.10
+= sΘb−1θpΘbθs
+by Lemma 4.10
+= pΘbθs = f
+by Lemma 4.10.
+The proof that e = fΘb��1θe is virtually identical.
+Lemmas 4.27 and 4.34 give the following:
+Theorem 4.36. If (G, ε) is a chained projection groupoid, then S(G, ε) is a regular ∗-semigroup.
+4.4
+Products of projections
+We conclude this chapter with some simple results concerning • products involving projections.
+These will be of use later, but we also apply them here to describe the idempotent-generated
+subsemigroup of the regular ∗-semigroup S = S(G, ε). In the following statements we continue
+to fix the chained projection groupoid (G, ε), and write P = vG.
+55
+
+Lemma 4.37. If a ∈ G(p, q), and if t ∈ P, then
+(i) a • t = a⇂q′ ◦ ε[q′, t′], where q′ = tθq and t′ = qθt,
+(ii) t • a = ε[t′, p′] ◦ p′⇃a, where p′ = tθp and t′ = pθt,
+(iii) a • t = a ◦ ε[q, t] if q F t,
+(iv) t • a = ε[t, p] ◦ a if p F t,
+(v) a • t = a⇂t if t ≤ q,
+(vi) t • a = t⇃a if t ≤ p.
+Proof. By symmetry, it suffices to prove (i), (iii) and (v).
+(i). We have a • t = a⇂q′ ◦ ε[q′, t′] ◦ t′⇃t = a⇂q′ ◦ ε[q′, t′] ◦ t′ = a⇂q′ ◦ ε[q′, t′].
+(iii). This follows from part (i), as q′ = q and t′ = t if q F t.
+(v). If t ≤ q, then we also have t ≤F q by Lemma 3.12(ii), so that t = tθq = q′ and t = qθt = t′.
+It then follows from part (i) that
+a • t = a⇂q′ ◦ ε[q′, t′] = a⇂t ◦ ε[t, t] = a⇂t ◦ t = a⇂t.
+Lemma 4.38. If p, q ∈ P, then
+(i) p • q = ε[p′, q′], where p′ = qθp and q′ = pθq,
+(ii) p • q = ε[p, q] if p F q.
+(iii) p • q • p = qθp.
+Proof. (i). By Lemma 4.37(i) we have p • q = p⇂p′ ◦ ε[p′, q′] = p′ ◦ ε[p′, q′] = ε[p′, q′].
+(ii). This follows from part (i), as p = p′ and q = q′ when p F q.
+(iii). Here we have
+p • q • p = ε[p′, q′] • p
+by part (i), where p′ = qθp and q′ = pθq
+= ε[p′, q′]⇂q′′ ◦ ε[q′′, p′′]
+by Lemma 4.37(i), where q′′ = pθq′ and p′′ = q′θp.
+Using the projection algebra axioms, it is easy to show that in fact p′′ = p′ and q′′ = q′. Thus,
+continuing from above, we have
+p • q • p = ε[p′, q′]⇂q′′ ◦ ε[q′′, p′′] = ε[p′, q′]⇂q′ ◦ ε[q′, p′] = ε[p′, q′] ◦ ε[q′, p′] = p′ = qθp.
+Recall that the sets of idempotents and projections of a regular ∗-semigroup S are denoted
+E(S) = {e ∈ S : e2 = e}
+and
+P(S) = {p ∈ S : p2 = p = p∗}.
+Although these sets are not equal in general, Lemma 2.23(ii) says that they generate the same
+subsemigroup of S, i.e. ⟨E(S)⟩ = ⟨P(S)⟩. We write E(S) for this idempotent-generated subsemi-
+group. The results proved above allow us to give the following description of E(S) in the case
+that S = S(G, ε) arises from a chained projection groupoid (G, ε).
+56
+
+Proposition 4.39. Let (G, ε) be a chained projection groupoid, and let P = vG, C = C (P) and
+S = S(G, ε). Then
+(i) P(S) = P,
+(ii) E(S) = {ε[p, q] : (p, q) ∈ F},
+(iii) E(S) = im(ε) = {ε(c) : c ∈ C }, and consequently S is idempotent-generated if and only
+if ε is surjective.
+Proof. (i). By Lemmas 2.23(i) and 4.26 we have
+P(S) = {a • a∗ : a ∈ S} = {a ◦ a−1 : a ∈ G} = {d(a) : a ∈ G} = vG = P.
+(ii). By Lemma 2.23(ii), and part (i), we have E(S) = {p • q : p, q ∈ P}. The claim then follows
+from Lemma 4.38.
+(iii). If c = [p1, . . . , pk] ∈ C , then
+ε(c) = ε[p1, p2] ◦ ε[p2, p3] ◦ · · · ◦ ε[pk−1, pk]
+by (4.13)
+= ε[p1, p2] • ε[p2, p3] • · · · • ε[pk−1, pk]
+by Lemma 4.26
+= (p1 • p2) • (p2 • p3) • · · · • (pk−1 • pk)
+by Lemma 4.38(ii)
+= p1 • p2 • · · · • pk ∈ E(S).
+Conversely, fix some a ∈ E(S), so that a = p1 • · · · • pk for some p1, . . . , pk ∈ P. We show
+that a ∈ im(ε) by induction on k. The k = 1 case being trivial, we assume that k ≥ 2, and let
+b = p1 • · · · • pk−1. By induction we have b = ε(c) for some c ∈ C , and we write r = r(b) = r(c).
+Then with r′ = pkθr and p′
+k = rθpk, it follows from Lemma 4.37(i) and properties of ε from
+Definition 4.11 that
+a = b • pk = b⇂r′ ◦ ε[r′, p′
+k] = ε(c)⇂r′ ◦ ε[r′, p′
+k] = ε(c⇂r′) ◦ ε[r′, p′
+k] = ε(c⇂r′ ◦ [r′, p′
+k]) ∈ im(ε).
+Remark 4.40. We will see in Chapter 6 (see Theorem 6.7) that every regular ∗-semigroup has
+the form S(G, ε) for some chained projection groupoid (G, ε).
+Combining this with Proposi-
+tion 4.39(iii), we obtain a proof of Proposition 2.42, stated in Chapter 2.
+5
+Regular ∗-semigroups
+In the previous chapter we showed that a chained projection groupoid (G, ε) gives rise to a
+regular ∗-semigroup S = S(G, ε); see Theorem 4.36. In the current chapter we go in the opposite
+direction. Starting from a regular ∗-semigroup S, we show how to construct a chained projection
+groupoid G(S) = (G, ε) from S; see Definitions 5.4 and 5.16, and Theorem 5.19. In the next
+chapter we will show that the S and G constructions are inverse processes, and in fact define
+isomorphisms between the categories of regular ∗-semigroups and chained projection groupoids.
+The construction of G(S) is given in Section 5.1. We then pause in Section 5.2 to consider
+a number of examples. These are intended to aid with understanding of the theory developed
+thus far, and also to help uncover some subtleties that arise.
+57
+
+5.1
+The chained projection groupoid associated to a regular ∗-semigroup
+For the duration of this chapter, we fix a regular ∗-semigroup S, as in Definition 2.20. As ever,
+we write
+P = P(S) = {p ∈ S : p2 = p = p∗}
+for the set of projections of S. As in Section 2.3, every projection p ∈ P gives rise to a map
+θp : P → P
+defined by
+qθp = pqp.
+(5.1)
+Comparing Lemma 2.25 with Definition 3.1, we immediately deduce the following:
+Proposition 5.2. If S is a regular ∗-semigroup, then P = P(S) is a projection algebra.
+Consequently, we have the partial order ≤ on P given in (3.5), and the relations ≤F and F
+on P given in (3.10) and (3.11). In particular (keeping (2.27) in mind), we note that
+p ≤ q
+⇔
+p = pθq = qpq
+⇔
+p = pq = qp
+⇔
+p = pq
+⇔
+p = qp.
+(5.3)
+We also have the path category P = P(P) and the chain groupoid C = C (P), as in Defini-
+tions 3.17 and 3.39.
+Recall that we wish to construct a chained projection groupoid G(S) = (G, ε) from S. The
+definition of the groupoid G = G(S) has already been given in Section 2.3; see Definition 2.32 and
+Proposition 2.33. Since we wish to show that G is in fact an ordered groupoid, it is convenient
+to give the following expanded definition.
+Definition 5.4. Given a regular ∗-semigroup S, we define the ordered groupoid G = G(S) as
+follows.
+• The object set of G is vG = P = P(S).
+• For a ∈ G we have d(a) = aa∗ and r(a) = a∗a.
+• For a, b ∈ G with r(a) = d(b), we have a ◦ b = ab.
+• For a ∈ G we have a−1 = a∗.
+The ordering on P = vG is given in (5.3), and left restrictions in G are defined by
+p⇃a = pa
+for p ≤ d(a).
+As usual, right restrictions are given by a⇂q = (q⇃a−1)−1, for q ≤ r(a), and in fact we have
+a⇂q = (qa∗)∗ = aq
+for such q. For a, b ∈ G, we have
+a ≤ b
+⇔
+a = p⇃b
+for some p ≤ d(b)
+⇔
+a = b⇂q
+for some q ≤ r(b)
+⇔
+a = p⇃b = b⇂q
+for some p ≤ d(b) and q ≤ r(b),
+and then of course p = d(a) and q = r(a).
+Lemma 5.5. If S is a regular ∗-semigroup, then G = G(S) is an ordered groupoid.
+58
+
+Proof. By Proposition 2.33, we just need to verify conditions (O1)′–(O5)′ from Lemma 2.3, with
+respect to the ordering (5.3), and the restrictions given in Definition 5.4.
+(O1)′. Consider a morphism a ∈ G, and let p ≤ d(a) = aa∗. It follows from p ≤ aa∗ that
+p = paa∗, and so
+d(p⇃a) = d(pa) = pa(pa)∗ = paa∗p∗ = pp = p.
+We also have
+r(p⇃a) = r(pa) = (pa)∗pa = a∗p∗pa = a∗pa.
+(5.6)
+It then follows that
+r(p⇃a) = a∗pa = a∗a · a∗pa · a∗a = r(a) · r(p⇃a) · r(a).
+By (5.3), the previous conclusion says precisely that r(p⇃a) ≤ r(a).
+(O2)′. By (5.6) we have q = a∗pa, and again p = paa∗ follows from p ≤ d(a). It follows that
+(p⇃a)∗ = (pa)∗ = a∗p∗ = a∗p = a∗paa∗ = qa∗ = q⇃a∗.
+(O3)′. Here d(a)⇃a = d(a)a = aa∗a = a.
+(O4)′. It follows from p ≤ q that p = pq, and so p⇃q⇃a = p(qa) = (pq)a = pa = p⇃a.
+(O5)′. Here we again have q = a∗pa. Since p ≤ d(a) we have p = aa∗p, so pa = aa∗pa = aq. It
+follows that
+p⇃(a ◦ b) = p(ab) = pp(ab) = paqb = p⇃a ◦ q⇃b.
+Remark 5.7. In particular, the relation ≤ on the regular ∗-semigroup S(= G) given in Defini-
+tion 5.4 is a partial order. This order was in fact used by Imaoka in [71], but in a different form.
+Imaoka’s order, which for clarity we will denote by ≤′, was defined by
+a ≤′ b
+⇔
+a ∈ Pb ∩ bP
+where P = P(S).
+(5.8)
+In light of the rules p⇃b = pb and b⇂q = bq, it is clear that a ≤ b ⇒ a ≤′ b. For the converse,
+suppose a ≤′ b, so that
+a = pb = bq
+for some p, q ∈ P.
+In the following calculations, we make extensive use of (5.3). We first note that
+a = bq
+⇒
+a = bb∗a
+⇒
+aa∗ = bb∗aa∗
+⇒
+aa∗ ≤ bb∗.
+We also have
+a = pb
+⇒
+a = pa
+⇒
+aa∗ = paa∗
+⇒
+aa∗ ≤ p
+⇒
+aa∗ = aa∗p.
+We then calculate
+a = aa∗a = aa∗(pb) = (aa∗p)b = aa∗b.
+Writing r = aa∗ ∈ P, we have already seen that r = aa∗ ≤ bb∗ = d(b), so that in fact
+a = aa∗b = rb = r⇃b, and hence a ≤ b.
+Remark 5.9. Recall that any regular semigroup S has a so-called natural partial order. This
+order, which for clarity we will denote by ⪯, has many different formulations; see for example
+[63,64,111], and especially [108] for a discussion of the variations, and an extension to arbitrary
+semigroups.
+The most straightforward definition of the natural partial order on the regular
+semigroup S is:
+a ⪯ b
+⇔
+a ∈ Eb ∩ bE
+where E = E(S).
+59
+
+It is natural to ask how the orders ≤ and ⪯ are related when S is a regular ∗-semigroup, where ≤
+is the order from Definition 5.4. Since P ⊆ E, and since ≤ is equal to ≤′ (cf. (5.8)), the order ≤
+is of course contained in ⪯, meaning that a ≤ b ⇒ a ⪯ b. But the converse does not hold,
+in general. For example, if S is a regular ∗-monoid with identity 1, and if e ∈ E \ P is any
+non-projection idempotent, then e ⪯ 1 but e ̸≤ 1.
+Next we wish to show that G is a projection groupoid, and to do this we need to understand
+the ϑ and Θ maps on G. For this, let a ∈ G. For p ≤ d(a) = aa∗, it follows from (5.6) that
+pϑa = r(p⇃a) = a∗pa.
+(5.10)
+It follows that for arbitrary p ∈ P,
+pΘa = pθd(a)ϑa = a∗(d(a) · p · d(a))a = a∗(aa∗ · p · aa∗)a = a∗pa.
+(5.11)
+In other words, ϑa and Θa both act by ‘conjugation’ on projections (cf. Remark 4.9), but we
+note that these maps have different domains: dom(ϑa) = d(a)↓ and dom(Θa) = P.
+Lemma 5.12. If S is a regular ∗-semigroup, then G = G(S) is a projection groupoid.
+Proof. By Lemma 5.5, it remains to check (G1). Specifically, we show that (G1a) holds, i.e. that
+θpϑa = Θa∗θpΘa
+for any a ∈ G and p ≤ d(a).
+But for any such a and p, and for arbitrary t ∈ P, we use (5.10) and (5.11) to calculate
+tθpϑa = tθa∗pa = a∗pa · t · a∗pa = tΘa∗θpΘa.
+Next we wish to construct an evaluation map ε = ε(S) : C → G, where C = C (P) is the
+chain groupoid of P = P(S). To do so, it is first convenient to define a map
+π = π(S) : P → G(= S)
+by
+π(p1, . . . , pk) = p1 · · · pk,
+(5.13)
+where P = P(P) is the path category of P. So π(p) is simply the product (in S) of the entries
+of p (taken in order). We begin with the following fact:
+Lemma 5.14. For any p ∈ P, we have d(π(p)) = d(p) and r(π(p)) = r(p).
+Proof. Write p = (p1, . . . , pk). A routine calculation shows that
+d(π(p)) = π(p) · π(p)∗ = p1 · · · pk−1 · pk · pk−1 · · · p1.
+Since p1 F · · · F pk, this reduces to p1 = d(p). We obtain r(π(p)) = pk = r(p) by symmetry.
+The next result concerns the congruence ≈ = Ω♯ on P from Definition 3.36.
+Lemma 5.15. The map π = π(S) : P → G is a v-functor, and ≈ ⊆ ker(π).
+Proof. For the first statement, it suffices by Lemma 5.14 to show that π is a functor. For this,
+consider composable paths p = (p1, . . . , pk) and q = (pk, . . . , pl). Then since pk is an idempotent,
+and since r(π(p)) = pk = d(π(q)) by Lemma 5.14, we have
+π(p ◦ q) = π(p1, . . . , pl) = p1 · · · pkpk+1 · · · pl
+= p1 · · · pk · pkpk+1 · · · pl = π(p)π(q) = π(p) ◦ π(q).
+To show that ≈ ⊆ ker(π), it suffices to show that Ω ⊆ ker(π), i.e. that π(s) = π(t) for all
+(s, t) ∈ Ω. This is clear if (s, t) has the form (Ω1) or (Ω2); for the latter, recall that p = qθp = pqp
+60
+
+when p F q. So suppose now that (s, t) has the form (Ω3). Consulting Definitions 3.36 and 3.31,
+this means that
+s = (e, eθp, f)
+and
+t = (e, fθp, f)
+for some p ∈ P, and some p-linked pair (e, f).
+Since ep, pf ∈ E(S) by Lemma 2.23(ii), it follows that
+π(s) = e · pep · f = epf = e · pfp · f = π(t).
+Definition 5.16. Since C = P/≈, it follows from Lemma 5.15 that we have a well-defined
+v-functor
+ε = ε(S) : C → G
+given by
+ε[p] = π(p)
+for p ∈ P.
+That is, ε[p1, . . . , pk] = p1 · · · pk whenever p1 F · · · F pk.
+Lemma 5.17. The functor ε = ε(S) : C → G is an evaluation map.
+Proof. Conditions (E1) and (E2) hold because ε is a v-functor. As explained in Remark 4.12,
+we can prove (E6) in place of (E3), and we note that (E6) says
+ε(q⇃[p]) = q⇃ε[p]
+for all p ∈ P and q ≤ d[p] = d(p).
+Since ε(q⇃[p]) = ε[q⇃p] = π(q⇃p) and q⇃ε[p] = q⇃π(p), this is equivalent to
+π(q⇃p) = q⇃π(p)
+for all p ∈ P and q ≤ d(p).
+(5.18)
+We prove this by induction on k, the length of the path p = (p1, . . . , pk). When k = 1, we have
+π(q⇃p) = π(q⇃(p1)) = π(q) = q = qp1 = qπ(p) = q⇃π(p),
+where we used the fact that q ≤ d(p) = p1
+⇒
+q = qp1.
+Now suppose k ≥ 2, and let
+q⇃p = (q1, . . . , qk) be as in (3.19). Let p′ = (p1, . . . , pk−1) ∈ P, noting that q⇃p′ = (q1, . . . , qk−1).
+Then
+q⇃π(p) = q · p1 · · · pk = qp1 · · · pk · (qp1 · · · pk)∗ · qp1 · · · pk
+= qp1 · · · pk · pk · · · p1q · qp1 · · · pk
+= q · (p1 · · · pk−1) · (pk · · · p1 · q · p1 · · · pk)
+= q⇃π(p′) · qθp1 · · · θpk
+= π(q⇃p′) · qk
+by induction and (3.20)
+= q1 · · · qk−1 · qk = π(q⇃p).
+This completes the proof of (5.18), and hence of the lemma.
+Note in particular that ε[p, q] = pq for p, q ∈ P with p F q. We are now ready to prove the
+main result of this chapter:
+Theorem 5.19. If S is a regular ∗-semigroup, then G(S) = (G, ε) is a chained projection
+groupoid.
+Proof. By Lemmas 5.12 and 5.17, it remains to verify (G2). To do so, let (e, f) be a b-linked
+pair, where b ∈ G(= S). Write
+q = d(b) = bb∗
+and
+r = r(b) = b∗b,
+and let the ei, fi be as in (4.19). Keeping (5.11) in mind, we have
+e1 = qeq,
+e2 = bfb∗,
+f1 = b∗eb
+and
+f2 = rfr.
+61
+
+We then calculate ee1 = eqeq = eq and bf1 = bb∗eb = qeb. Since e ≤F q by Lemma 4.18(i), we
+also have e = qθe = eqe. Putting this all together, we have
+λ(e, b, f) = ε[e, e1] ◦ e1⇃b ◦ ε[f1, f] = ee1 · e1b · f1f = ee1 · bf1 · f = eq · qeb · f = eqe · bf = ebf.
+A similar calculation gives ρ(e, b, f) = ebf, and the proof is complete.
+Remark 5.20. In the above proof, verification of (G2) boils down to checking that
+ee1bf1f = ee2bf2f,
+where b ∈ G(q, r), (e, f) is b-linked, and the ei, fi are as in (4.19). We are now in a position to see
+that condition (G2) cannot be replaced by the stronger (G2)′ discussed in Remark 4.23. Indeed,
+consider the partition monoid P4, and the elements defined by
+b =
+,
+e = e1 = f1 =
+,
+e2 = f2 =
+,
+f =
+.
+These elements are all projections (so in particular q = r = b), and they satisfy conditions (i)–(iv)
+of Lemma 4.18. However, we have
+ee1bf1f =
+̸=
+= ee2bf2f.
+5.2
+Examples
+At this point it is worth pausing to consider a number of examples in a fair amount of detail.
+These are included to illustrate the general theory developed above, but also to demonstrate
+some subtleties.
+Our first example shows that it is possible for two non-isomorphic regular ∗-semigroups S1
+and S2 to have exactly the same ordered groupoids G(S1) = G(S2).
+Example 5.21. Let P be an arbitrary set, and to avoid trivialities we assume that |P| ≥ 2. We
+will define two semigroups S1 and S2, both with underlying set (P × P) ∪ {0}. To distinguish
+the products, we will denote them by ⋆1 and ⋆2. The element 0 will act as a zero element in
+both semigroups, and the rest of the products are defined by
+(p, q) ⋆1 (r, s) = (p, s)
+and
+(p, q) ⋆2 (r, s) =
+�
+(p, s)
+if q = r
+0
+otherwise.
+So these are in fact special cases of some basic semigroup constructions:
+• S1 is a P × P square band with a zero attached, and
+• S2 is a P × P combinatorial Brandt semigroup.
+Both S1 and S2 are regular ∗-semigroups with respect to the involution
+(p, q)∗ = (q, p)
+and
+0∗ = 0.
+In fact, S2 is inverse (as a Brandt semigroup), while S1 is not (as the idempotents (p, p) and
+(q, q) do not commute, for distinct p, q ∈ P). In particular, S1 and S2 are not isomorphic; this
+also follows from the fact that S1 has no non-trivial zero divisors, while every element of S2 is a
+zero divisor.
+62
+
+The regular ∗-semigroups S1 and S2 have the same projection sets, and the projections are
+precisely 0 and the pairs (p, p), for p ∈ P. So, identifying (p, p) ≡ p, we have
+P(S1) = P(S2) ≡ P ∪ {0}.
+On the other hand, the idempotents are the products of two projections (cf. Lemma 2.23(ii)).
+Of course any product involving 0 is 0, while for p, q ∈ P we have
+p ⋆1 q = (p, q)
+and
+p ⋆2 q =
+�
+p
+if p = q
+0
+otherwise.
+This shows that
+E(S1) = S1
+and
+E(S2) = P(S2) = P ∪ {0}.
+(Of course these facts are also easy to see directly.) This difference is also reflected at the level
+of projection algebras. In what follows, we use θi to denote the projection algebra operations
+on Si (i = 1, 2), as in (5.1). Of course θ1
+0 = θ2
+0 is the constant map with image {0}, while for
+p ∈ P we have
+xθ1
+p =
+�
+p
+if x ∈ P
+0
+if x = 0
+and
+xθ2
+p =
+�
+p
+if x = p
+0
+if x = 0 or x ∈ P \ {p}.
+(5.22)
+Thus, the projection algebras P1 = P(S1) and P2 = P(S2) are different, despite having the same
+underlying sets. It is also easy to see that the F relations on P1 and P2 are very different.
+Indeed, denoting these by F1 and F2, we have
+F1 = ∇P ∪ {(0, 0)}
+and
+F2 = ∆P∪{0}.
+On the other hand, the partial orders on P1 and P2 (see (5.3)) are the same. These orders have
+the following Hasse diagram, shown in the case P = {p1, . . . , p5}:
+p1
+p2
+p3
+p4
+p5
+0
+(5.23)
+It turns out that the groupoids G1 = G(S1) and G2 = G(S2) are exactly the same as well. Indeed,
+to see this, we note that the object sets are equal: vG1 = vG2 = P ∪ {0}. Next we note that
+Green’s R and L relations are the same on S1 and S2. Indeed, in both semigroups we have
+R0 = L0 = {0}, and
+(p, q) R (r, s)
+⇔
+p = r
+and
+(p, q) L (r, s)
+⇔
+q = s.
+It then follows that the non-empty morphism sets in Gi (i = 1, 2) are
+Gi(0, 0) = {0}
+and
+Gi(p, q) = Rp ∩ Lq = {(p, q)}
+for p, q ∈ P.
+The operations in the groupoids Gi (i = 1, 2) are given by
+(p, q) ◦ (q, r) = (p, r)
+and
+(p, q)−1 = (p, q)∗ = (q, p).
+Moreover, the order in both groupoids are the same. Indeed, keeping the order on P1 = P2
+in mind (cf. (5.23)), the morphism 0 is below every other morphism, and there are no other
+non-trivial relations.
+63
+
+We have just seen that it is possible for two non-isomorphic regular ∗-semigroups S1 ̸∼= S2
+to have exactly the same ordered groupoids, G(S1) = G(S2).
+It follows from this that the
+groupoid alone is not enough information to distinguish regular ∗-semigroups; in other words,
+the groupoid is not a total invariant of a regular ∗-semigroup.
+However, the semigroups S1
+and S2 in Example 5.21 had different projection algebra structures.
+Thus, even though the
+groupoids G1 and G2 are identical (in terms of their compositions, inversions and orderings), we
+can still distinguish them by means of the underlying projection algebra structure of their object
+sets, vG1 ̸∼= vG2.
+This may seem like a strange point, as the projection algebra structure is somewhat ‘invisible’
+in the structure of the groupoid G. In a sense, the projection algebra is merely used as a means
+to define the order on vG, which then feeds in to the order on G itself. However, G is not only
+an ordered groupoid, but in fact a projection groupoid (cf. Definition 4.8), and this means that
+there is a stronger link between the structures of G and vG than merely the order-theoretic one
+just described. Indeed, the defining properties (G1a)–(G1d) from Definition 4.8 involve the Θa
+maps from (4.1), which are defined in terms of the order-theoretic properties of G (specifically,
+the ϑa maps) and the projection algebra structure of vG (specifically, the θp maps). And it is
+easy to see that the groupoids G1 = G(S1) and G2 = G(S2) have different Θ maps. Indeed, again
+distinguishing these by using the symbols Θ1 and Θ2, we even have Θ1
+p = θ1
+p ̸= θ2
+p = Θ2
+p for
+p ∈ P; cf. (4.2) and (5.22).
+In any case, one may wonder if the pair (G, P) consisting of the groupoid G = G(S) and the
+projection algebra P = P(S), including its structure as determined by the θ operations, might
+be a total invariant for a regular ∗-semigroup. The next example shows that this is still not the
+case, and illustrates the need for the evaluation map in the chained projection groupoid (G, ε).
+Example 5.24. Fix a group G, and an arbitrary set P. Also let M = (mpq)p,q∈P be a P × P
+‘sandwich matrix’ over G with the property that
+mqp = m−1
+pq
+for all p, q ∈ P.
+(5.25)
+Note that (5.25) forces mpp = 1 for all p ∈ P, where here 1 denotes the identity of the group G.
+The Rees matrix semigroup SM = M (G, P, M) has underlying set P × G × P, and product
+(p, g, q)(r, h, s) = (p, gmqrh, s).
+One can easily check (using (5.25)) that SM becomes a regular ∗-semigroup under the involution
+given by
+(p, g, q)∗ = (q, g−1, p).
+The projections of SM are the elements (p, 1, p) for each p ∈ P, and we identify p ≡ (p, 1, p).
+With this identification, one can easily check that pqp = p for all p, q ∈ P, so that each θp map
+(as in (5.1)) is the constant map with image {p}. It follows that the projection algebra structure
+of P(SM) is independent of the sandwich matrix M, and depends only on the set P. Note also
+that the law p = pqp = qθp gives p ≤F q for all p, q ∈ P, which means that F = ∇P is the
+universal relation on P. On the other hand, the order ≤ on P is trivial, as p ≤ q ⇔ p = pθq = q.
+As ever, the idempotents of SM are the products of two projections:
+pq ≡ (p, 1, p)(q, 1, q) = (p, mpq, q)
+for p, q ∈ P.
+In particular, the idempotents of SM depend on the sandwich matrix M.
+We now consider the groupoid G = G(SM). (The reason for not including an M-subscript
+on G will become clear shortly.) As ever, the object set of G is vG = P. Green’s R and L
+relations on SM are given by
+(p, g, q) R (r, h, s)
+⇔
+p = r
+and
+(p, g, q) L (r, h, s)
+⇔
+q = s.
+64
+
+In particular, the morphism sets of G are given by
+G(p, q) = Rp ∩ Lq = {p} × G × {q} = {(p, g, q) : g ∈ G}
+for each p, q ∈ P.
+The operations in the groupoid G are given by
+(p, g, q) ◦ (q, h, r) = (p, gh, r)
+and
+(p, g, q)−1 = (p, g, q)∗ = (q, g−1, p).
+Note that the above rule for composition in G holds because of the identity mqq = 1 (cf. (5.25)).
+In particular, these operations are independent of the sandwich matrix M. Since the ordering
+on P is trivial, so too is the ordering on G.
+Thus, given distinct matrices M1 and M2 satisfying (5.25), the semigroups S1 = SM1 and
+S2 = SM2 will give rise to identical projection algebras P(S1) = P(S2), including the θ operations,
+and identical groupoids G(S1) = G(S2).
+However, the semigroups S1 and S2 are of course
+different, as they will have different products (since M1 ̸= M2). Moreover, S1 and S2 need not
+even be isomorphic. Indeed, if every entry of M1 is 1 (the identity of G), then S1 is isomorphic
+to the direct product of G with the square band P ×P, and in particular the idempotents E(S1)
+form a subsemigroup of S1. But this need not be the case in general; indeed, if |G| ≥ 2 and
+|P| ≥ 3, and if we choose the entries of M2 = (mpq) in such a way that mpq = mqr = 1 ̸= mpr
+for distinct p, q, r ∈ P, then (p, 1, q) and (q, 1, r) are both idempotents of S2, but their product
+(p, 1, q)(q, 1, r) = (p, 1, r) ̸= (p, mpr, r)
+is not. Thus, S1 ̸∼= S2 in this case.
+It follows from this that the pair (G, P) consisting of the groupoid G = G(S) and projection
+algebra P = P(S) is not a total invariant of a regular ∗-semigroup. It turns out that the missing
+ingredient is the evaluation map ε = ε(S) from Definition 5.16. Indeed, we will see that the
+pair (G, ε), i.e. the chained projection groupoid associated to S, is a total invariant.
+Going back to the Rees matrix semigroup SM, we now consider the evaluation map
+εM = ε(SM) : C → G,
+where here C = C (P) is the chain groupoid of P. As in (4.13), and keeping F = ∇P in mind, εM
+is determined entirely by the elements εM[p, q], for p, q ∈ P. Following Definition 5.16, we have
+εM[p, q] = pq ≡ (p, mpq, q).
+In particular, the evaluation map εM depends on the sandwich matrix M.
+In a sense, the previous conclusion is reversible. Indeed, if we start from the pair (G, P),
+consisting of the groupoid G, and the projection algebra P (with each θp being the constant map
+with image {p}), we have immense freedom in defining an evaluation map ε : C → G. Indeed, we
+can do so by defining the elements ε[p, q], for p, q ∈ P, to be arbitrary morphisms from G(p, q), as
+long as we ensure that ε[q, p] = ε[p, q]−1 for all p, q. Since ε[p, q] must be of the form (p, mpq, q)
+for some mpq ∈ G, and since then ε[p, q]−1 = (q, m−1
+pq , p), this condition on inverses is of course
+equivalent to (5.25). Thus, roughly speaking, evaluation maps C → G and sandwich matrices
+satisfying (5.25) are one and the same thing.
+On the other hand, the idempotent structure of SM, as determined by the biordered set E(SM),
+is independent of M. The biordered set of a semigroup S consists of the set E(S) of idempotents,
+together with a partial binary operation (e, f) �→ ef (a restriction of the product in S), defined
+precisely when ef and/or fe is equal to one of e or f. These ‘basic products’ can be conveniently
+depicted using Easdown’s arrow notation from [31]; we write
+e
+f
+or
+e
+f
+65
+
+to indicate that ef = e (e is a left zero for f) or fe = e (e is a right zero for f), respectively. In
+the case that S = SM, we have seen that E = E(S) consists of the elements epq = (p, mpq, q),
+for p, q ∈ P. One can easily check that
+epq
+ers
+⇔
+p = r,
+epq
+ers
+⇔
+q = s.
+The entire biordered structure of E(S) is then as follows, in the case P = {1, 2, 3, 4}, and
+remembering that the
+and
+relations are transitive:
+e11
+e21
+e31
+e41
+e12
+e22
+e32
+e42
+e13
+e23
+e33
+e43
+e14
+e24
+e34
+e44
+6
+The category isomorphism
+We are now ready to bring all of the above ideas together. The previous two chapters gave
+constructions for:
+• building a regular ∗-semigroup S(G, ε) from a chained projection groupoid (G, ε), and
+• building a chained projection groupoid G(S) from a regular ∗-semigroup S.
+As we have already discussed, one of the goals of the current chapter is to show that the S and G
+constructions are inverse processes, meaning that
+G(S(G, ε)) = (G, ε)
+and
+S(G(S)) = S
+for any chained projection groupoid (G, ε), and any regular ∗-semigroup S. In fact, we go further
+than this, and show in Theorem 6.7 below that S and G are mutually inverse isomorphisms
+between the categories of regular ∗-semigroups and chained projection groupoids.
+Let RSS be the (large) category of all regular ∗-semigroups. Morphisms in RSS are the
+∗-semigroup homomorphisms, i.e. the maps
+φ : S1 → S2
+such that
+(ab)φ = (aφ)(bφ)
+and
+(a∗)φ = (aφ)∗
+for all a, b ∈ S1.
+We also define CPG to be the (large) category of all chained projection groupoids. Mor-
+phisms in CPG are what we call chained projection functors:
+66
+
+Definition 6.1. Consider two chained projection groupoids (G1, ε1) and (G2, ε2). For i = 1, 2,
+we write Pi = vGi, Ci = C (Pi), and so on. A chained projection functor from (G1, ε1) to (G2, ε2)
+is an ordered functor φ : G1 → G2 satisfying the following two conditions:
+(F1) The object map vφ : P1 → P2 (i.e. the restriction of φ to P1 = vG1) is a projection algebra
+morphism, meaning that
+(pθq)φ = (pφ)θqφ
+for all p, q ∈ P1.
+(Here we use θ to denote the projection algebra operations on both P1 and P2.)
+(F2) The functor φ respects the evaluation maps, meaning that the following diagram commutes:
+C1
+C2
+G1
+G2,
+Φ
+ε1
+ε2
+φ
+where here Φ : C1 → C2 is the induced functor given by [p1, . . . , pk]Φ = [p1φ, . . . , pkφ], as
+in Proposition 3.41. Commutativity of the diagram means that
+(ε1(c))φ = ε2(cΦ)
+for all c ∈ C1.
+The main result of this chapter, Theorem 6.7 below, states that the categories RSS and
+CPG are isomorphic. We build towards the theorem with a series of lemmas.
+Lemma 6.2. If S1 and S2 are regular ∗-semigroups, then any ∗-semigroup homomorphism
+S1 → S2 is a chained projection functor G(S1) → G(S2).
+Proof. Fix a ∗-semigroup homomorphism φ : S1 → S2, and write G(Si) = (Gi, εi), Pi = P(Si) = vGi,
+Ci = C (Pi), and so on. Since the underlying sets of G1 and G2 are S1 and S2, we can think of φ
+as a map G1 → G2. Since φ is a ∗-semigroup homomorphism, it follows that for a ∈ G1 we have
+d(aφ) = (aφ)(aφ)∗ = (aφ)(a∗φ) = (aa∗)φ = d(a)φ
+and similarly
+r(aφ) = r(a)φ.
+But then if a, b ∈ G1 are such that r(a) = d(b), then r(aφ) = d(bφ), so we have
+(a ◦ b)φ = (ab)φ = (aφ)(bφ) = (aφ) ◦ (bφ).
+Thus, φ is a functor.
+Before we check that φ is order-preserving, we note that (F1) holds, as for any p, q ∈ P1,
+(pθq)φ = (qpq)φ = (qφ)(pφ)(qφ) = (pφ)θqφ.
+To see that φ preserves the order, suppose a, b ∈ G1 are such that a ≤ b. Then a = p⇃b = pb
+for some p ≤ d(b). Note that
+p ≤ d(b)
+⇒
+p = pθd(b)
+⇒
+pφ = (pφ)θd(b)φ
+⇒
+pφ ≤ d(b)φ = d(bφ).
+It then follows that
+aφ = (pb)φ = (pφ)(bφ) = pφ⇃(bφ) ≤ bφ.
+It remains to check that (F2) holds. For this, let c = [p1, . . . , pk] ∈ C1. Then
+(ε1(c))φ = (p1 · · · pk)φ = (p1φ) · · · (pkφ) = ε2[p1φ, . . . , pkφ] = ε2(cΦ).
+67
+
+Lemma 6.3. If (G1, ε1) and (G2, ε2) are chained projection groupoids, then any chained projection
+functor (G1, ε1) → (G2, ε2) is a ∗-semigroup homomorphism S(G1, ε1) → S(G2, ε2).
+Proof. Fix a chained projection functor φ : G1 → G2. Also write S1 = S(G1, ε1) and S2 = S(G2, ε2).
+Again we can think of φ as a map S1 → S2. Since φ is a groupoid functor, we have
+(a∗)φ = (a−1)φ = (aφ)−1 = (aφ)∗
+for all a ∈ S1.
+To make the rest of this proof notationally concise, we will write a = aφ for a ∈ S1. It remains
+to show that φ is a semigroup homomorphism, i.e. that a • b = a • b for all a, b ∈ S1. (Here we
+write • for the product on both S1 and S2.) For this, write as usual
+p = r(a),
+q = d(b),
+p′ = qθp
+and
+q′ = pθq,
+so that
+a • b = a⇂p′ ◦ ε1[p′, q′] ◦ q′⇃b.
+Since φ is a functor, we have
+r(a) = r(a ◦ p) = r(a ◦ p) = r(p) = p
+and similarly
+d(b) = q.
+Thus,
+a • b = a⇂p′ ◦ ε2[p′, q′] ◦ q′⇃b
+where
+p′ = qθp
+and
+q′ = pθq.
+By (F1) we have
+p′ = qθp = qθp = p′
+and similarly
+q′ = q′.
+By (F2) we have
+ε1[p′, q′] = (ε1[p′, q′])φ = ε2([p′, q′]Φ) = ε2[p′, q′] = ε2[p′, q′].
+Since φ is an ordered functor, we have
+a⇂p′ = a⇂p′ = a⇂p′
+and similarly
+q′⇃b = q′⇃b.
+Putting everything together, we have
+a • b = a⇂p′ ◦ ε1[p′, q′] ◦ q′⇃b = a⇂p′ ◦ ε1[p′, q′] ◦ q′⇃b = a⇂p′ ◦ ε2[p′, q′] ◦ q′⇃b = a • b.
+Lemma 6.4. If (G, ε) is a chained projection groupoid, then G(S(G, ε)) = (G, ε).
+Proof. Write
+S = S(G, ε)
+and
+(G′, ε′) = G(S).
+We must show that G′ = G and ε′ = ε. By the equality G′ = G, we mean that these have
+the same underlying sets, and all the same operations and relations (composition, inversion,
+order and projection algebra operations). The underlying sets of S and G′ are simply G, and
+the unary operations of S and G′ are simply inversion in G. We also have vG = P(S) = vG′
+(cf. Proposition 4.39(i)), and we denote this set of projections by P.
+Recall that the product • in S is given by Definition 4.24. The groupoid G′ is constructed
+from S, as outlined in Definition 5.4. To avoid ambiguity, we will denote the composition on G′
+by ⊚ in order to distinguish it from the original composition ◦ in G. For the same reason, we
+also denote the domain and range maps in G′ by d′ and r′. So for a, b ∈ G′(= G), we have
+d′(a) = a • a−1,
+r′(a) = a−1 • a
+and
+a ⊚ b = a • b when r′(a) = d′(b).
+By Lemma 4.26, and since r(a) = d(a−1), it follows that
+d′(a) = a • a−1 = a ◦ a−1 = d(a)
+and similarly
+r′(a) = r(a).
+(6.5)
+68
+
+It also follows that
+r′(a) = d′(b)
+⇔
+r(a) = d(b)
+and then
+a ⊚ b = a • b = a ◦ b.
+Thus, the compositions in the categories G and G′ coincide.
+Next we consider the projection algebra operations. We denote those in G as usual by θp
+(p ∈ P), and those in G′ by θ′
+p (p ∈ P), remembering that vG′ = vG = P. We must show that
+qθp = qθ′
+p for all p, q ∈ P, and this in fact follows from Lemma 4.38(iii):
+qθ′
+p = p • q • p = qθp.
+Since θp = θ′
+p for all p ∈ P = vG = vG′, it follows that p ≤ q in G precisely when p ≤ q in G′.
+We now move on to the orders on G and G′. We use the symbols ⇃ and ⇂ to denote restrictions
+in G as usual, and ↿ and ↾ for restrictions in G′ (note the placement of the arrow heads). Let
+a ∈ G and write q = d(a). By (6.5), p⇃a is defined in G precisely when p↿a is defined in G′,
+i.e. when p ≤ q. And for such p, it follows from Definition 5.4 and Lemma 4.37(vi) that
+p↿a = p • a = p⇃a.
+But then for a, b ∈ G, and writing ≤ and ≤′ for the orders in G and G′, we have
+a ≤ b
+⇔
+a = p⇃b for some p ≤ d(b)
+⇔
+a = p↿b for some p ≤ d′(b)
+⇔
+a ≤′ b.
+All of the above shows that the ordered groupoids G and G′ coincide. It remains to show that
+ε = ε′. To do so, fix some c = [p1, . . . , pk] ∈ C . By Definition 5.16 we have
+ε′(c) = p1 • · · · • pk,
+and we show that ε(c) = ε′(c) by induction on k. The k = 1 case being clear, suppose k ≥ 2,
+and write d = [p1, . . . , pk−1] and a = ε(d). Note that c = d ◦ [pk−1, pk], and by induction we have
+a = ε(d) = ε′(d) = p1 • · · · • pk−1.
+It follows that ε′(c) = a•pk. Since r(a) = r(d) = pk−1 (as ε′ is a v-functor), and since pk−1 F pk
+(as c ∈ C ), it follows from Lemma 4.37(iii) that
+ε′(c) = a • pk = a ◦ ε[pk−1, pk] = ε(d) ◦ ε[pk−1, pk] = ε(d ◦ [pk−1, pk]) = ε(c),
+as required.
+Lemma 6.6. If S is a regular ∗-semigroup, then S(G(S)) = S.
+Proof. Write
+(G, ε) = G(S)
+and
+S′ = S(G, ε).
+Again, by construction the underlying sets of S and S′ coincide, and so do the involutions. It
+therefore remains to show that the products in S and S′ coincide as well. We denote the product
+in S by juxtaposition, as usual. The product in S′ is •, which is constructed from (G, ε) as in
+Definition 4.24, while (G, ε) is in turn constructed from S as in Definitions 5.4 and 5.16.
+So let a, b ∈ S; we must show that a • b = ab. Write p = r(a) and q = d(b), so that
+a • b = a⇂p′ ◦ ε[p′, q′] ◦ q′⇃b
+where
+p′ = qθp
+and
+q′ = pθq.
+Following the definitions in Chapter 5, and using Lemma 2.23, we have
+p = a∗a,
+p′ = pqp,
+a⇂p′ = ap′ = apqp,
+q = bb∗,
+q′ = qpq,
+q′⇃b = q′b = qpqb,
+ε[p′, q′] = p′q′ = pqpqpq = pq.
+Combining all of the above, and keeping Definition 5.4 and Lemma 2.23 in mind, it follows that
+a • b = apqp ◦ pq ◦ qpqb = apqpqpqb = apqb = aa∗abb∗b = ab,
+as required.
+69
+
+We now have all we need to prove the main result.
+Theorem 6.7. The category RSS of regular ∗-semigroups (with ∗-semigroup homomorphisms)
+is isomorphic to the category CPG of chained projection groupoids (with chained projection
+functors).
+Proof. By Theorems 4.36 and 5.19 we have maps
+S : CPG → RSS : (G, ε) �→ S(G, ε)
+and
+G : RSS → CPG : S �→ G(S).
+By Lemmas 6.2 and 6.3, S and G are functors, and by Lemmas 6.4 and 6.6 they are mutually
+inverse.
+Remark 6.8. We can use Theorem 6.7 to deduce a structure theorem for regular ∗-semigroups, in
+the sense that it shows how the entire structure of such a semigroup can be entirely, and uniquely,
+determined by ‘simpler’ objects. Specifically, given a regular ∗-semigroup S, one constructs:
+• the projection groupoid G = G(S), as in Definition 5.4, including its object set vG = P(S),
+which is the projection algebra of S, and
+• the evaluation map ε = ε(S), as in Definition 5.16.
+Lemma 6.6 tells us that S is completely determined by these two simpler objects, in the sense
+that S = S(G, ε) can be constructed from G and ε in the manner described in Definition 4.24.
+Examples 5.21 and 5.24 demonstrate the subtlety of the decomposition, showing how seemingly-
+small changes to G and/or ε can result in very different semigroups.
+Theorem 6.7 shows that the categories of regular ∗-semigroups and chained projection groupoids
+are in a sense the same. It also follows from the definition of the functors S and G that the
+algebras of projections of regular ∗-semigroups are precisely the object sets of chained projection
+groupoids. However, we are still left with the following two (equivalent) questions, which we will
+explore in some depth in the second part of the paper:
+Question 6.9. Given an (abstract) projection algebra P, as in Definition 3.1:
+• does there exist a regular ∗-semigroup S with P(S) = P?
+• does there exist a chained projection groupoid (G, ε) with vG = P?
+Question 6.9 (or at least the first version of it) is already known to have an affirmative
+answer.
+For example, Imaoka [70, 72] and Yamada [138] have both shown how to construct
+a maximum fundamental regular ∗-semigroup with a given projection algebra P (definitions
+are given below), though Yamada worked with a somewhat different formulation of projection
+algebras. This fundamental semigroup was also studied by Jones in [73, Section 4], but he was
+not concerned with maximality. As an application of our results, we will give another (ultimately
+equivalent) construction of this maximum fundamental semigroup in Chapter 8. We also answer
+Question 6.9 by an alternative route in Chapter 7, by constructing a free (projection-generated)
+regular ∗-semigroup with projection algebra P. As the name suggests, these semigroups have
+many remarkable (categorical) ‘free-ness’ properties, which will be elaborated upon further in
+Chapter 7, and can be constructed in a variety of ways, including by a presentation by generators
+and relations (see Theorem 7.14).
+70
+
+Part II
+Applications
+The first part of this paper was devoted to a proof of our main structural result, Theorem 6.7,
+which established the isomorphism of the categories of regular ∗-semigroups and chained projec-
+tion groupoids. In this second part, we explore a number of consequences of this theorem:
+• Chapter 7 uses chain groupoids to construct the free regular ∗-semigroup over an arbitrary
+projection algebra; see Theorems 7.11, 7.14, 7.20 and 7.23.
+• Chapter 8 uses projection groupoids to give a new, transparent, construction of fundamental
+regular ∗-semigroups; see Theorems 8.20, 8.21 and Theorem 8.24.
+• Chapter 9 applies our theory to the special case of inverse semigroups, leading to a new proof
+of the celebrated Ehresmann–Schein–Nambooripad Theorem, stated in Theorem 9.1.
+Again, the introduction of each chapter contains a detailed outline of its structure, and a summary
+of the main results.
+7
+Idempotent-generated regular ∗-semigroups
+As discussed at the end of Chapter 6, one of our motivations at this point is to show how to
+construct, from an arbitrary projection algebra P (cf. Definition 3.1), a regular ∗-semigroup S
+with projection algebra P(S) = P or, equivalently, a chained projection groupoid (G, ε) with
+object set vG = P. As it happens, we have already constructed such a groupoid, namely the chain
+groupoid C = C (P) from Definition 3.39, with evaluation map simply the identity idC , although
+we still have to verify quite a few non-trivial details. The chained projection groupoid (C , idC )
+leads as usual to the regular ∗-semigroup S(C , idC ), which we will call the chain semigroup of P,
+and denote by CP . As we will see, this semigroup possesses many important properties, beyond
+simply being a regular ∗-semigroup with projection algebra P(CP ) = P.
+In Section 7.1 we prove Theorem 7.11, which shows that CP is universal among idempotent-
+generated regular ∗-semigroups in the sense that:
+• CP is an idempotent-generated regular ∗-semigroup with projection algebra P, and
+• any idempotent-generated regular ∗-semigroup with projection algebra P is a homomorphic
+image of CP .
+(Recall from Lemma 2.23(ii) that a regular ∗-semigroup is idempotent-generated if and only if it
+is projection-generated.) As explained in Remark 7.12, this shows that CP is a free/initial object
+in the category IGRSSP of idempotent-generated regular ∗-semigroups with (fixed) projection
+algebra P. In Section 7.2 we give a presentation for CP in terms of generators and relations;
+see Theorem 7.14.
+We then return in Section 7.3 to our discussion of the ‘free-ness’ of CP .
+Theorem 7.20 demonstrates another universal property of CP among all regular ∗-semigroups:
+• For any projection algebra P and regular ∗-semigroup S, any projection algebra morphism
+φ : P → P(S) extends uniquely to a ∗-semigroup morphism CP → S.
+This is then used to prove Theorem 7.23, which says that CP is the free regular ∗-semigroup
+with projection algebra P, in the sense that the semigroups CP are precisely the objects in the
+image of a left adjoint of a suitable forgetful functor. (The relevant definitions are given below.)
+We then conclude the chapter with some open questions.
+71
+
+7.1
+The chain semigroup
+For the duration of this section we fix a projection algebra P, as in Definition 3.1, and all of
+the data that comes with it, i.e. the unary operations θp, the relations ≤, ≤F and F, and so
+on. We also write P = P(P) and C = C (P) for the path category and chain groupoid of P;
+cf. Definitions 3.17 and 3.39.
+Our first goal is to show that C is a projection groupoid, for which it remains to ver-
+ify (G1), specifically (G1a). To do so, we need to understand the maps ϑc and Θc, for a P-chain
+c = [p1, . . . , pk] ∈ C .
+Keeping d(c) = p1 and r(c) = pk in mind, and using (3.20), the map
+ϑc : p↓
+1 → p↓
+k is given by
+qϑc = r(q⇃c) = qθp2 · · · θpk = qθp1 · · · θpk
+for q ≤ p1.
+(7.1)
+It follows from this that qΘc = qθd(c)ϑc = qθp1θp2 · · · θpk for arbitrary q ∈ P. This all shows that
+Θc = θp1 · · · θpk
+for c = [p1, . . . , pk] ∈ C .
+(7.2)
+Lemma 7.3. For any projection algebra P, the chain groupoid C = C (P) is a projection
+groupoid.
+Proof. To verify (G1a), consider a P-chain c = [p1, . . . , pk] ∈ C , and let q ≤ d(c) = p1. We
+must show that
+θqϑc = Θc−1θqΘc.
+For this we use (7.1), Lemma 3.4 and (7.2) to calculate
+θqϑc = θqθp1···θpk = θpk · · · θp1θqθp1 · · · θpk = Θc−1θqΘc,
+where in the last step we recall that c−1 = [pk, . . . , p1].
+Next, it is clear that the identity map ε = idC is an evaluation map (cf. Definition 4.11).
+Proposition 7.4. For any projection algebra P, (C , idC ) is a chained projection groupoid, where
+C = C (P) is the chain groupoid of P.
+Proof. It remains to verify (G2). To do so, fix a c-linked pair (e, f), where c = [p] ∈ C for
+p = (p1, . . . , pk) ∈ P. Following Definitions 4.16, 4.20 and 4.22 this means that
+f = eΘcθf
+and
+e = fΘc−1θe,
+(7.5)
+and we must show that λ(e, c, f) = ρ(e, c, f). Keeping ε = idC in mind, these morphisms are
+defined by
+λ(e, c, f) = [e, e1] ◦ e1⇃c ◦ [f1, f]
+and
+ρ(e, c, f) = [e, e2] ◦ c⇂f2 ◦ [f2, f],
+(7.6)
+in terms of the projections
+e1 = eθp1,
+e2 = fΘc−1,
+f1 = eΘc
+and
+f2 = fθpk.
+For convenience, we will write
+e1⇃c = [u1, . . . , uk]
+and
+c⇂f2 = [v1, . . . , vk].
+Using (3.19) and (3.30), and remembering that e1 = eθp1 and f2 = fθpk, we have
+ui = e1θp2 · · · θpi = eθp1 · · · θpi
+and
+vi = f2θpk−1 · · · θpi = fθpk · · · θpi,
+72
+
+for each 1 ≤ i ≤ k. Keeping in mind that the compositions in (7.6) exist, it follows from all this
+that
+λ(e, c, f) = [e, u1, . . . , uk, f]
+and
+ρ(e, c, f) = [e, v1, . . . , vk, f],
+and we must show that these chains are equal. It will also be convenient to additionally write
+u0 = e and vk+1 = f. To assist with understanding the coming arguments, these projections are
+shown in Figure 16 (in the case k = 4). Our task is essentially to show that the large rectangle
+at the bottom of the diagram commutes, modulo ≈.
+Next we claim that (ui−1, vi+1) is pi-linked for each 1 ≤ i ≤ k, as in Definition 3.31. To prove
+this, we must show that
+vi+1 = ui−1θpiθvi+1
+and
+ui−1 = vi+1θpiθui−1.
+First we note that (7.2) and (7.5) give
+f = eθp1 · · · θpkθf
+and
+e = fθpk · · · θp1θe.
+Combining this with Lemma 3.4, we obtain
+ui−1θpiθvi+1 = eθp1 · · · θpi−1 · θpi · θfθpk···θpi+1
+= eθp1 · · · θpi−1 · θpi · θpi+1 · · · θpkθfθpk · · · θpi+1 = fθpk · · · θpi+1 = vi+1.
+The proof that ui−1 = vi+1θpiθui−1 is essentially identical. Now that we have proved the claim,
+these linked pairs lead to the paths
+λ(ui−1, pi, vi+1) = (ui−1, ui−1θpi, vi+1)
+and
+ρ(ui−1, pi, vi+1) = (ui−1, vi+1θpi, vi+1)
+= (ui−1, ui, vi+1)
+= (ui−1, vi, vi+1),
+as in Definition 3.31. It then follows by the form of Ω in Definition 3.36 (see (Ω3)) that
+[ui−1, vi, vi+1] = [ui−1, ui, vi+1]
+for each 1 ≤ i ≤ k.
+(7.7)
+In other words, each of the small rectangles at the bottom of Figure 16 commute. It follows,
+therefore, that the large rectangle commutes. Formally, we repeatedly use (7.7) in the indicated
+places, as follows:
+ρ(e, c, f) = [e, v1, . . . , vk, f] = [u0, v1, v2, v3, v4, . . . , vk, vk+1]
+= [u0, u1, v2, v3, v4, . . . , vk, vk+1]
+= [u0, u1, u2, v3, v4, . . . , vk, vk+1]
+...
+= [u0, u1, u2, u3, u4, . . . , uk, vk+1] = [e, u1, . . . , uk, f] = λ(e, c, f),
+and the proof is complete.
+As a result of Proposition 7.4 and Theorem 4.36, we have a regular ∗-semigroup S(C , idC ),
+which we will denote by CP . To give an explicit description of the • product in CP , consider
+an arbitrary pair of P-chains c = [p1, . . . , pk] and d = [q1, . . . , ql]. Also write p = r(c) = pk and
+q = d(d) = q1. Following Definition 4.24, and remembering that the evaluation map is idC , we
+have
+c • d = c⇂p′ ◦ [p′, q′] ◦ q′⇃d
+where
+p′ = qθp
+and
+q′ = pθq.
+As in (3.19) and (3.30), we have c⇂p′ = [p′
+1, . . . , p′
+k] and q′⇃d = [q′
+1, . . . , q′
+l], where
+p′
+i = p′θpk · · · θpi
+and
+q′
+j = q′θq1 · · · θqj
+for 1 ≤ i ≤ k and 1 ≤ j ≤ l,
+73
+
+e = u0
+v5 = f
+p1
+u1
+v1
+p2
+u2
+v2
+p3
+u3
+v3
+p4
+u4
+v4
+Figure 16. The projections e, f, pi, ui, vi from the proof of Proposition 7.4, shown here in the case
+k = 4. Dashed lines indicate ≤ relationships. Each arrow s → t represents the P-path (s, t) ∈ P, so
+the upper and lower paths e → f represent (e, u1, . . . , uk, f) and (e, v1, . . . , vk, f), respectively.
+and where p′ = p′
+k and q′ = q′
+1. Then
+c • d = c⇂p′ ◦ [p′, q′] ◦ q′⇃d = [p′
+1, . . . , p′
+k] ◦ [p′
+k, q′
+1] ◦ [q′
+1, . . . , q′
+l] = [p′
+1, . . . , p′
+k, q′
+1, . . . , q′
+l]
+is simply the concatenation of c⇂p′ and q′⇃d. We denote the concatenation of a, b ∈ C by a ⊕ b,
+but we note that this only belongs to C if r(a) F d(b). As special cases we have
+c • d = c ⊕ d if r(c) F d(d)
+and
+c • d = c ◦ d if r(c) = d(d).
+Definition 7.8. Given a projection algebra P (cf. Definition 3.1), we define the regular ∗-
+semigroup
+CP = S(C , idC ),
+where C = C (P) is the chain groupoid of P (cf. Definition 3.39). Explicitly:
+• The elements of CP are the P-chains, [p1, . . . , pk], as in Definition 3.39.
+• The product • in CP is given, for c, d ∈ CP with r(c) = p and d(d) = q, by
+c • d = c⇂p′ ⊕ q′⇃d
+where
+p′ = qθp
+and
+q′ = pθq,
+and where ⊕ denotes concatenation, as above.
+• The involution in CP is given by reversal of P-chains, [p1, . . . , pk]∗ = [pk, . . . , p1].
+• The projections of CP are the trivial chains [p] ≡ p, for p ∈ P, and consequently P(CP ) ≡ P.
+• The idempotents of CP are the chains [p, q], for (p, q) ∈ F.
+We call CP the chain semigroup of P.
+For the coming results of this chapter, and for later use, we require the following definition:
+Definition 7.9. Let S and S′ be regular ∗-semigroups with the same projection algebras,
+P(S) = P(S′) = P.
+A ∗-semigroup homomorphism φ : S → S′ is said to be canonical if
+φ|P = idP , i.e. if pφ = p for all p ∈ P.
+74
+
+Proposition 7.10. For any chained projection groupoid (G, ε), the evaluation map ε is a chained
+projection functor (C , idC ) → (G, ε), and (consequently) a canonical ∗-semigroup homomorphism
+CP → S(G, ε).
+Proof. By Lemma 6.3 (and since the v-functor ε : C → G has object map vε = idP ), it suffices
+to prove the first claim. Since ε is an ordered v-functor C → G (cf. Definition 4.11), we just need
+to check that conditions (F1) and (F2) both hold. The first follows immediately from vε = idP .
+For (F2), vε = idP means that the induced map Φ : C → C at the top of the diagram in
+Definition 6.1 is the identity map. Thus, this diagram is
+C
+C
+C
+G,
+idC
+idC
+ε
+ε
+which obviously commutes.
+Theorem 7.11. If P is a projection algebra, then
+(i) CP is an idempotent-generated regular ∗-semigroup with projection algebra P,
+(ii) any idempotent-generated regular ∗-semigroup with projection algebra P is a canonical im-
+age of CP .
+Proof. (i).
+By Theorem 4.36, CP = S(C , idC ) is a regular ∗-semigroup.
+It is idempotent-
+generated by Proposition 4.39(iii), since the evaluation map idC : C → C is obviously surjective.
+We have already observed that P(CP ) ≡ P.
+(ii). Let S be an idempotent-generated regular ∗-semigroup with projection algebra P. By Propo-
+sition 7.10 (and Theorem 6.7), ε is a canonical ∗-semigroup homomorphism CP → S(G(S)) = S,
+and by Proposition 4.39(iii), it is surjective.
+Remark 7.12. We now briefly comment on a categorical interpretation of Theorem 7.11. For a
+fixed projection algebra P, we denote by IGRSSP the (typically large) category of idempotent-
+generated regular ∗-semigroups with projection algebra P. The morphisms in IGRSSP are the
+canonical ∗-homomorphisms from Definition 7.9. Since every object in IGRSSP is generated
+by P (cf. Lemma 2.23(ii)), and since every morphism in IGRSSP maps P identically, there can
+be at most one such morphism S → S′, for any pair of objects S, S′ of IGRSSP , which must
+then be surjective.
+Theorem 7.11 essentially states that CP is a free/initial object in IGRSSP . We will say
+more about the free-ness of CP (in the full category RSS) in Section 7.3; see in particular
+Theorems 7.20 and 7.23.
+Before we move on, we record the following simple consequence of our previous results.
+Proposition 7.13. Any projection algebra morphism φ : P → P ′ extends to a (unique) well-
+defined ∗-semigroup homomorphism
+Φ : CP → CP ′
+given by
+[p1, . . . , pk]Φ = [p1φ, . . . , pkφ].
+Proof. By Proposition 3.41, the mapping Φ is an ordered groupoid functor C → C ′, where as
+usual we write C = C (P) and C ′ = C (P ′). By Lemma 6.3, it suffices to show that Φ is in
+fact a chained projection functor (C , idC ) → (C ′, idC ′), as it will then also be a ∗-semigroup
+75
+
+homomorphism CP = S(C , idC ) → S(C ′, idC ′) = CP ′. Consulting Definition 6.1, we are left to
+show that Φ satisfies conditions (F1) and (F2).
+(F1). By definition we have vΦ = φ, and this is a projection algebra morphism by assumption.
+(F2). Keeping in mind that the evaluation maps are the identities, the diagram in Definition 6.1
+becomes
+C
+C ′
+C
+C ′,
+Φ
+idC
+idC′
+Φ
+and this obviously commutes.
+7.2
+Presentation by generators and relations
+Our next result is a presentation by generators and relations for the chain semigroup CP . Roughly
+speaking, we take an abstract copy of P as a generating set, and impose the bare minimum of
+relations to ensure that the quotient of the free semigroup on P is a regular ∗-semigroup with
+projection algebra P. This idea is akin to the constructions of Pastijn [115] and Easdown [31] of
+the free (regular) idempotent-generated semigroup over an arbitrary (regular) biordered set E.
+They both take a semigroup defined by a presentation with generating set (an abstract copy
+of) E, and relations coming from the ‘basic products’, and the ‘sandwich sets’ in the regular
+case.
+Before we state the result, we briefly establish some notation. For a set X, we denote by X+
+the free semigroup over X, which consists of all non-empty words over X, under concatenation.
+For a set R ⊆ X+ × X+ of pairs of words, we write R♯ for the congruence on X+ generated
+by R. We say a semigroup S has presentation ⟨X : R⟩ if S ∼= X+/R♯, i.e. if there is a surjective
+semigroup homomorphism X+ → S with kernel R♯. At times we identify ⟨X : R⟩ with the semi-
+group X+/R♯ itself. The elements of X and R are called generators and relations, respectively.
+A relation (u, v) ∈ R is typically displayed as an equation: u = v.
+Theorem 7.14. For any projection algebra P, the chain semigroup CP has presentation
+CP ∼= ⟨XP : RP ⟩,
+where XP = {xp : p ∈ P} is an alphabet in one-one correspondence with P, and where RP is the
+set of relations
+x2
+p = xp
+for all p ∈ P,
+(R1)
+(xpxq)2 = xpxq
+for all p, q ∈ P,
+(R2)
+xpxqxp = xqθp
+for all p, q ∈ P.
+(R3)
+To prove the theorem, we require some technical lemmas. But first, it is worth observing that
+relations (R1)–(R3) closely resemble projection algebra axioms (P2), (P4) and (P5), i.e. those
+that are stated purely in terms of the θ maps.
+For the rest of this section, we fix the projection algebra P, and also the alphabet XP and
+relations RP from Theorem 7.14. We also write ∼ = R♯
+P for the congruence on X+
+P generated
+by relations (R1)–(R3). For words u, v ∈ X+
+P , we write u ∼1 v to indicate that u and v differ by
+one or more applications of (R1), and similarly for ∼2 and ∼3.
+76
+
+Lemma 7.15. If p1, . . . , pk, q ∈ P are such that p1 F · · · F pk and q ≤ pk, then
+xp1 · · · xpk−1xq ∼ xq1 · · · xqk−1xq
+for some q1, . . . , qk−1 ∈ P with q1 F · · · F qk−1 F q.
+Proof. The proof is by induction on k. When k = 1 the result is vacuously true, so we as-
+sume k ≥ 2. Then with qk−1 = qθpk−1, we have
+xpk−1xq ∼2 xpk−1xqxpk−1xq ∼3 xqθpk−1xq = xqk−1xq.
+By induction, noting that qk−1 ≤ pk−1, we have
+xp1 · · · xpk−2xqk−1 ∼ xq1 · · · xqk−2xqk−1
+for some q1, . . . , qk−2 ∈ P with q1 F · · · F qk−2 F qk−1. It follows that
+xp1 · · · xpk−2xpk−1xq ∼ xp1 · · · xpk−2xqk−1xq ∼ xq1 · · · xqk−2xqk−1xq,
+and it remains only to show that qk−1 F q.
+But from q ≤ pk F pk−1 it follows from
+Lemma 3.15(ii) that q F qθpk−1 = qk−1.
+Lemma 7.16. For any w ∈ X+
+P of length k, we have
+w ∼ xp1 · · · xpk
+for some p1, . . . , pk ∈ P with p1 F · · · F pk.
+Proof. We prove the lemma by induction on k. The k = 1 case being trivial, we assume k ≥ 2,
+and we write w = xq1 · · · xqk. By induction we have
+xq1 · · · xqk−1 ∼ xr1 · · · xrk−1
+for some r1, . . . , rk−1 ∈ P with r1 F · · · F rk−1. Next we define
+pk−1 = qkθrk−1
+and
+pk = rk−1θqk,
+noting that pk−1 F pk, by Lemma 3.14. We then calculate
+xrk−1xqk ∼2 xrk−1xqkxrk−1xqkxrk−1xqk ∼3 xqkθrk−1xrk−1θqk = xpk−1xpk.
+Since pk−1 ≤ rk−1, it follows from Lemma 7.15 that
+xr1 · · · xrk−2xpk−1 ∼ xp1 · · · xpk−2xpk−1
+for some p1, . . . , pk−2 ∈ P with p1 F · · · F pk−2 F pk−1. Putting everything together, we have
+w = xq1 · · · xqk−2xqk−1xqk ∼ xr1 · · · xrk−2xrk−1xqk ∼ xr1 · · · xrk−2xpk−1xpk ∼ xp1 · · · xpk−2xpk−1xpk,
+and the proof is complete.
+Given a P-path p = (p1, . . . , pk) ∈ P = P(P), we define the word
+wp = xp1 · · · xpk ∈ X+
+P .
+So Lemma 7.16 says that every word over XP is ∼-equivalent to some wp. Using (R1), it is easy
+to see that
+wpwq ∼ wp◦q
+for any p, q ∈ P with r(p) = d(q).
+(7.17)
+(In fact, we have wpwq = wp⊕q when r(p) F d(q), where ⊕ again denotes the concatenation
+operation.) The next result refers to the congruence ≈ on P from Definition 3.36.
+77
+
+Lemma 7.18. For any p, q ∈ P, we have p ≈ q ⇒ wp ∼ wq.
+Proof. It suffices to assume that p and q differ by a single application of (Ω1)–(Ω3), i.e. that
+p = p′ ◦ s ◦ p′′
+and
+q = p′ ◦ t ◦ p′′
+for some p′, p′′ ∈ P and (s, t) ∈ Ω ∪ Ω−1.
+Since wp ∼ wp′wswp′′ and wq ∼ wp′wtwp′′, by (7.17), it is in fact enough to prove that
+ws ∼ wt
+for all (s, t) ∈ Ω.
+We consider the three forms the pair (s, t) ∈ Ω can take.
+(Ω1). This follows immediately from (R1).
+(Ω2). If s = (p, q, p) and t = (p) for some (p, q) ∈ F, then ws = xpxqxp ∼3 xqθp = xp = wt.
+(Ω3). Finally, suppose s = λ(e, p, f) = (e, eθp, f) and t = ρ(e, p, f) = (e, fθp, f) for some p ∈ P,
+and some p-linked pair (e, f). Then
+ws = xexeθpxf ∼3 xexpxexpxf ∼2 xexpxf ∼2 xexpxfxpxf ∼3 xexfθpxf = wt.
+We can now tie together the loose ends.
+Proof of Theorem 7.14. Define the homomorphism
+Ψ : X+
+P → CP
+by
+xpΨ = p ≡ [p]
+for p ∈ P.
+To see that Ψ is surjective, let c ∈ CP , so that c = [p] for some p = (p1, . . . , pk) ∈ P. Then as
+in the proof of Proposition 4.39(iii), and remembering that CP = S(C , idC ), we have
+c = idC (c) = p1 • · · · • pk = (xp1 · · · xpk)Ψ = wpΨ.
+(7.19)
+Next, we note that Ψ preserves the relations RP , meaning that
+(u, v) ∈ RP
+⇒
+uΨ = vΨ (in CP ).
+Indeed, this follows immediately from Lemma 2.23(ii) when (u, v) has type (R1) or (R2), and
+from Lemma 4.38(iii) for type (R3). It follows from this that R♯
+P ⊆ ker(Ψ).
+It remains to show that ker(Ψ) ⊆ R♯
+P . To do so, fix some (u, v) ∈ ker(Ψ), so that u, v ∈ X+
+P
+and uΨ = vΨ; we must show that u ∼ v. By Lemma 7.16, we have u ∼ wp and v ∼ wq for
+some p, q ∈ P. Using (7.19), and remembering that ∼ ⊆ ker(Ψ), we have
+[p] = wpΨ = uΨ = vΨ = wqΨ = [q],
+meaning that p ≈ q. But then wp ∼ wq by Lemma 7.18, so u ∼ wp ∼ wq ∼ v, as required.
+In what follows, we will typically denote the R♯
+P -class of a word w ∈ X+
+P by [w]. In this way,
+we have
+⟨XP : RP ⟩ = X+
+P /R♯
+P = {[w] : w ∈ X+
+P }.
+The product in ⟨XP : RP ⟩ is given by [u][v] = [uv] for u, v ∈ X+
+P , and the involution by
+[xp1 · · · xpk]∗ = [xpk · · · xp1] for p1, . . . , pk ∈ P. The projections of ⟨XP : RP ⟩ are the R♯
+P -classes
+[xp] ≡ p (p ∈ P), so the projection algebra of ⟨XP : RP ⟩ is (isomorphic to) P.
+78
+
+7.3
+Chain semigroups as free objects
+In Remark 7.12 we discussed a certain ‘free-ness’ property of the chain semigroup CP in the
+category of idempotent-generated regular ∗-semigroups with (fixed) projection algebra P. In
+category theory, ‘free’ objects are more formally defined as the objects in the image of a ‘left
+adjoint to a forgetful functor’. We will soon recall the meaning of these terms, and show that
+our chain semigroups fit this definition, with respect to an appropriate forgetful functor.
+We begin by proving the following result.
+It will be used to verify the above-mentioned
+categorical condition, but we hope that the reader can already sense a ‘flavour’ of free-ness from
+the statement, and the universal property it describes.
+Theorem 7.20. If P is a projection algebra and S is a regular ∗-semigroup, then for any projec-
+tion algebra morphism φ : P → P(S), there is a unique ∗-semigroup homomorphism Φ : CP → S
+such that the following diagram commutes (where both ‘ι’s denote inclusion maps):
+P
+P(S)
+CP
+S.
+φ
+ι
+ι
+Φ
+Proof. This could be proved by constructing a suitable chained projection functor (C , idC ) → G(S),
+where C = C (P) is the chain groupoid, and then applying Lemma 6.3. Alternatively, we can
+use Theorem 7.14, and prove the result with ⟨XP : RP ⟩ = X+
+P /R♯
+P in place of CP . Taking this
+second route, we begin by defining a semigroup homomorphism
+ϕ : X+
+P → S
+by
+xpϕ = pφ
+for p ∈ P.
+Next we check that RP ⊆ ker(ϕ), meaning that uϕ = vϕ for all (u, v) ∈ RP . Indeed, this is
+essentially trivial (modulo Lemma 2.23(ii)) when (u, v) has type (R1) or (R2). For type (R3),
+we must show that (xpxqxp)ϕ = xqθpϕ for any p, q ∈ P. But for any such p, q, and using θ′ to
+denote the unary operations in P(S), as in (5.1), we have
+(xpxqxp)ϕ = (pφ)(qφ)(pφ) = (qφ)θ′
+pφ = (qθp)φ = xqθpϕ.
+(In the penultimate step we used the fact that φ is a projection algebra morphism.)
+It follows from RP ⊆ ker(ϕ) that R♯
+P ⊆ ker(ϕ). We therefore have a well-defined semigroup
+homomorphism
+Φ : ⟨XP : RP ⟩ → S
+given by
+[w]Φ = wϕ
+for w ∈ X+
+P .
+This is a ∗-homomorphism, as for any word w = xp1 · · · xpk ∈ X+
+P ,
+[w]∗Φ = [xpk · · · xp1]Φ = (pkφ) · · · (p1φ) = (pkφ)∗ · · · (p1φ)∗ = ((p1φ) · · · (pkφ))∗ = ([w]Φ)∗.
+Commutativity of the diagram amounts to the fact that Φ|P = φ (where as above we identify
+p ≡ [xp] for p ∈ P). This also implies uniqueness, as CP is generated by P ≡ {[xp] : p ∈ P}.
+Consider again the two (large) categories:
+• RSS, of regular ∗-semigroups with ∗-semigroup homomorphisms, and
+• PA, of projection algebras with projection algebra morphisms.
+79
+
+The process of taking the projection algebra P(S) of a regular ∗-semigroup S can be thought of
+as a ‘forgetful functor’; we forget the non-projections of S, and remember only the ‘conjugation’
+actions of projections on each other. Guided by standard letter usage (see for example [96, p. 79]),
+we denote this functor by
+U : RSS → PA.
+So the action of U on an object S ∈ vRSS and morphism φ ∈ RSS(S, S′) is given by:
+• U(S) = P(S) is the projection algebra of S, with unary operations as in (5.1), and
+• U(φ) : U(S) → U(S′) is the (set-theoretic) restriction of φ to U(S), which is easily seen to be
+a projection algebra morphism.
+In the other direction, the process of constructing the chain semigroup CP of a projection alge-
+bra P is a functor
+F : PA → RSS.
+The action of F on an object P ∈ vPA and morphism φ ∈ PA(P, P ′) is given by:
+• F(P) = CP is the chain semigroup of P, as in Definition 3.39, and
+• F(φ) : F(P) → F(P ′) is the induced ∗-semigroup homomorphism Φ : CP → CP ′ from
+Proposition 7.13.
+So we have the two functors
+U : RSS → PA
+and
+F : PA → RSS.
+We have already observed that U(F(P)) = P(CP ) = P for any projection algebra P. It is
+also clear that U(F(φ)) = φ for any morphism φ in PA (this is precisely what it means for Φ to
+extend φ in Proposition 7.13). In other words, UF = idPA is the identity functor PA → PA. The
+other composition FU : RSS → RSS is not the identity. However, we will show in Theorem 7.23
+that F is a left adjoint to U.
+To make the previous statement precise, we need to recall some definitions. There are many
+equivalent definitions of adjunctions, but for our purposes the most convenient is that of [5,
+Definition 9.1].
+Definition 7.21. Consider two (possibly large) categories C and D, and a pair of functors
+F, G : C → D. A natural transformation η : F → G is a family η = (ηC)C∈vC, where each
+ηC : F(C) → G(C) is a morphism in D, and such that the following condition holds:
+• For every pair of objects C, C′ ∈ vC, and for every morphism φ : C → C′ in C, the following
+diagram commutes:
+F(C)
+G(C)
+F(C′)
+G(C′).
+ηC
+F(φ)
+G(φ)
+ηC′
+80
+
+Definition 7.22. Consider two (possibly large) categories C and D. An adjunction C → D is a
+triple (F, U, η), where F : C → D and U : D → C are functors, and η is a natural transformation
+idC → UF, such that the following condition holds:
+• For every pair of objects C ∈ vC and D ∈ vD, and for every morphism φ : C → U(D), there
+exists a unique morphism φ : F(C) → D such that the following diagram commutes:
+C
+U(F(C))
+U(D).
+φ
+ηC
+U(φ)
+In this set-up, F and U are called the left and right adjoints, respectively, and η is the unit of
+the adjunction. The C-free objects in D are the objects in the image of F, i.e. those of the form
+F(C) for C ∈ vC.
+Theorem 7.23. The functor
+F : PA → RSS : P �→ CP
+is a left adjoint to the forgetful functor
+U : RSS → PA : S �→ P(S).
+Consequently, the U-free objects in the category RSS are precisely the chain semigroups.
+Proof. Consulting Definition 7.22, we need a natural transformation η : idPA → UF. Since we
+have already observed that UF = idPA, we can take η = (idP )P∈vPA. For P, P ′ ∈ vPA and
+φ : P → P ′, the diagram in Definition 7.21 becomes
+P
+P
+P ′
+P ′,
+idP
+φ
+φ
+idP ′
+and this obviously commutes.
+To verify that (F, U, η) is an adjunction PA → RSS, we need to show that:
+• For every projection algebra P ∈ vPA and regular ∗-semigroup S ∈ vRSS, and for every
+morphism φ : P → P(S), there exists a unique ∗-semigroup homomorphism φ : CP → S such
+that the following diagram commutes:
+P
+P
+P(S).
+φ
+idP
+φ|P
+For such P, S and φ, we take φ = Φ : CP → S, as in Theorem 7.20, and the required properties
+(uniqueness and commutativity) follow from the theorem.
+81
+
+It follows from Theorem 7.23 that we may rightly speak of CP as ‘the free regular ∗-semigroup
+with projection algebra P’.
+There is a vast literature on free idempotent-generated semigroups FIG(E) over biordered
+sets E; see for example [11, 17, 19, 23–25, 27–32, 56–58, 110, 112, 140]. A long-standing folklore
+conjecture was that the maximal subgroups of any FIG(E) are all free groups. The first coun-
+terexample was constructed in [11], and then the conjecture was proven to be maximally false
+in [57], where it was shown that every group appears as the maximal subgroup of some free
+idempotent-generated semigroup; this result has been reproved a number of times [25, 56, 140].
+Since then, a number of important studies have computed maximal subgroups arising from
+biordered sets of natural families of semigroups, such as transformation monoids [58] and linear
+monoids [23]. We believe it would be very interesting to study the free (idempotent/projection-
+generated) regular ∗-semigroups CP along similar lines. Natural questions include the following:
+Problem 7.24.
+(i) Are maximal subgroups of CP always free?
+(ii) Can any group appear as the maximal subgroup of some CP ?
+(iii) What are the maximal subgroups of CP , when P is the projection algebra of some natural
+family of regular ∗-semigroups (e.g. partition, Brauer or Temperley-Lieb monoids)?
+(iv) Given a regular ∗-semigroup S, with projection algebra P = P(S) and biordered set
+E = E(S), how are the free semigroups CP and FIG(E) related? Since E is a regular
+biordered set, the same question can be asked of the free regular idempotent-generated
+semigroup FRIG(E); cf. [115].
+Some of these questions will be explored in [37]. We will say a little about the last question in
+Example 7.25 below. One could also study structural properties of CP , or consider decision prob-
+lems, as in [17,19,24]. The above papers typically study free idempotent-generated semigroups
+via Easdown’s presentation [31]. To study CP , one has the option of using its presentation from
+Theorem 7.14 or its direct definition as the chain semigroup of P.
+Example 7.25. Let P be an arbitrary set, and consider again the square band over P, i.e. the
+regular ∗-semigroup S = P × P, with operations
+(p, q)(r, s) = (p, s)
+and
+(p, q)∗ = (q, p).
+The projections of S are of the form (p, p), for p ∈ P.
+Identifying (p, p) ≡ p, we see that
+P(S) ≡ P. For each p ∈ P, the operation θp is the constant map with image {p}, and F = ∇P
+is the universal relation.
+In the special case that P = {p, q} has size 2, the chain semigroup
+CP = {[p], [q], [p, q], [q, p]}
+has size 4, and of course CP ∼= S. This can also be seen by writing down the presentation from
+Theorem 7.14, which simplifies (writing xp ≡ p and xq ≡ q) to
+⟨p, q : p2 = p, q2 = q, pqp = p, qpq = q⟩.
+It is clear that {p, q, pq, qp} is a set of normal forms, i.e. a set of representatives of equivalence-
+classes of words.
+On the other hand, the biordered set E = E(S) of S is simply E = S, and it is easy to
+see that FIG(E) is infinite; indeed, simple computations in GAP [52, 107] show that maximal
+subgroups of FIG(E) are infinite cyclic. This all shows that the answer to the fourth question
+in Problem 7.24 will not simply be that CP and FIG(E) are always isomorphic.
+82
+
+8
+Fundamental regular ∗-semigroups
+In this chapter we turn our attention to fundamental regular ∗-semigroups. Many of the ideas
+discussed here exist in the literature in a different form [70, 72, 73, 138], but we feel that our
+groupoid approach leads to some clearer constructions and proofs.
+We begin in Section 8.1 by obtaining a number of formulations for the maximum projection-
+separating congruence µS on a regular ∗-semigroup S; see Proposition 8.3 and Corollary 8.4.
+In Section 8.2 we construct the maximum fundamental regular ∗-semigroup MP with a given
+projection algebra P; see Definition 8.19. Theorems 8.20 and 8.21 establish various universal
+properties of the semigroup MP . Specifically, we use the ϑ maps on chained projection groupoids
+to show that for any regular ∗-semigroup S with projection algebra P(S) = P, there is a canonical
+∗-semigroup homomorphism φS : S → MP , and that im(φS) ∼= S/µS is the fundamental image
+of S. In particular, if S happens to be fundamental itself, then φS is an embedding of S in MP .
+At times we compare our construction of MP with others from the literature. In Section 8.3 we
+consider idempotent-generated fundamental regular ∗-semigroups. In particular, Theorem 8.24
+shows that (up to isomorphism) there is exactly one such semigroup with a given projection
+algebra, and gives a number of ways to construct it.
+8.1
+The maximum idempotent-separating congruence
+Recall that a congruence σ on a semigroup S is idempotent-separating if
+e σ f
+⇒
+e = f
+for all e, f ∈ E(S).
+A ∗-congruence on a regular ∗-semigroup S is a congruence σ that is also compatible with ∗:
+a σ b
+⇒
+a∗ σ b∗
+for all a, b ∈ S.
+We say that such a σ is projection-separating if
+p σ q
+⇒
+p = q
+for all p, q ∈ P(S).
+Any such projection-separating ∗-congruence σ is idempotent-separating, as for e, f ∈ E(S),
+e σ f ⇒ e∗ σ f∗ ⇒ ee∗ σ ff∗ ⇒ ee∗ = ff∗ ⇒ e R f
+and similarly
+e L f.
+It follows that e σ f ⇒ e H f ⇒ e = f, since an H -class can contain at most one idempotent.
+Definition 8.1. A regular ∗-semigroup S is fundamental if it has no non-trivial idempotent-
+separating (equivalently, projection-separating) ∗-congruences, i.e. if the only idempotent-separating
+∗-congruence is the trivial relation ∆S = {(a, a) : a ∈ S}.
+Given a regular ∗-semigroup S, it was shown in [70, Theorem 4] and [138, Theorem 2.4] that
+the relation
+µS = {(a, b) ∈ S × S : a∗pa = b∗pb and apa∗ = bpb∗ for all p ∈ P}
+(8.2)
+is the maximum projection-separating ∗-congruence on S, and also the maximum idempotent-
+separating (ordinary semigroup) congruence on S. It follows that a regular ∗-semigroup is fun-
+damental as a ∗-semigroup (as in Definition 8.1) if and only if it is fundamental as a semigroup
+(i.e. has no non-trivial idempotent-separating (semigroup) congruences). It also follows that S/µS
+is the unique fundamental quotient of S with the same projection algebra (up to isomorphism);
+we call S/µS the fundamental image of S.
+To keep the paper self-contained, it is worth briefly sketching why (8.2) holds, i.e. why the
+stated relation is the maximum projection-separating congruence:
+83
+
+• First, it is easy to check that µS is a ∗-congruence.
+• It is projection-separating because for p, q ∈ P,
+(p, q) ∈ µS ⇒ p = p∗pp = q∗pq = qpq ⇒ p ≤ q
+and similarly
+q ≤ p,
+which shows that (p, q) ∈ µS ⇒ p = q.
+• It is the maximum such, because if σ is an arbitrary projection-separating ∗-congruence, then
+for any a, b ∈ S and p ∈ P,
+a σ b ⇒ a∗ σ b∗ ⇒ a∗pa σ b∗pb ⇒ a∗pa = b∗pb
+and similarly
+apa∗ = bpb∗,
+which shows that (a, b) ∈ σ ⇒ (a, b) ∈ µS, i.e. that σ ⊆ µS.
+It turns out that the congruence µS has a useful alternative description, in terms of our ϑ and Θ
+maps:
+Proposition 8.3. If S is a regular ∗-semigroup, then the maximum idempotent-separating con-
+gruence of S is the relation
+µS = {(a, b) ∈ S × S : ϑa = ϑb} = {(a, b) ∈ S × S : Θa = Θb and Θa∗ = Θb∗}
+= {(a, b) ∈ S × S : d(a) = d(b) and Θa = Θb}.
+Proof. It follows immediately from (5.11) that Θa = Θb
+⇔
+a∗pa = b∗pb for all p ∈ P.
+Combining this with the definition of µS in (8.2), we have
+µS = {(a, b) ∈ S × S : Θa = Θb and Θa∗ = Θb∗}.
+It therefore remains to show that the following are equivalent, for all a, b ∈ S:
+(i) ϑa = ϑb
+(ii) Θa = Θb and Θa∗ = Θb∗,
+(iii) d(a) = d(b) and Θa = Θb.
+Equivalence of (i) and (iii) follows quickly from the fact that ϑa = Θa|d(a)↓ and Θa = θd(a)ϑa.
+(i) ⇒ (ii). Suppose ϑa = ϑb. Considering the domains of these mappings (see (2.10)), it follows
+that d(a) = d(b), and then
+Θa = θd(a)ϑa = θd(b)ϑb = Θb.
+By Lemma 2.12, we also have ϑa∗ = ϑ−1
+a
+= ϑ−1
+b
+= ϑb∗. But then the previous argument also
+gives Θa∗ = Θb∗.
+(ii) ⇒ (iii).
+Now suppose Θa = Θb and Θa∗ = Θb∗.
+Considering the range of Θa∗ = Θb∗
+(see (4.1)), it follows that d(a) = d(b).
+This also leads to a somewhat simpler equational characterisation of µS, as compared to the
+original in (8.2). For another equivalent formulation, see for example [114, Corollary 4.6].
+Corollary 8.4. If S is a regular ∗-semigroup, then the maximum idempotent-separating congru-
+ence of S is the relation
+µS = {(a, b) ∈ S × S : aa∗ = bb∗, a∗pa = b∗pb for all p ≤ aa∗}
+= {(a, b) ∈ S × S : a∗a = b∗b, apa∗ = bpb∗ for all p ≤ a∗a}.
+84
+
+Proof. By Proposition 8.3 we have (a, b) ∈ µS ⇔ ϑa = ϑb. This is of course equivalent to the
+following two conditions:
+(i) dom(ϑa) = dom(ϑb), and
+(ii) pϑa = pϑb for all p ∈ dom(ϑa).
+Since dom(ϑa) = (aa∗)↓ and dom(ϑb) = (bb∗)↓, (i) is equivalent to aa∗ = bb∗. Combining this
+with (5.10), it follows that (ii) is equivalent to a∗pa = b∗pb for all p ≤ aa∗.
+This gives the first claimed expression for µS. The second follows from the first because µS
+is a ∗-congruence, or from the fact that ϑa = ϑb ⇔ ϑa∗ = ϑb∗ (cf. Lemma 2.12).
+8.2
+Maximum fundamental regular ∗-semigroups
+Coming from the other direction, it has also long been known that for any projection algebra P,
+there is a unique ‘maximum’ fundamental regular ∗-semigroup with projection algebra P (up to
+isomorphism). This maximum semigroup has been constructed in a number of ways; the papers
+[70,72,73,113,138] start from projection algebras (or similar structures), while [113] also contains
+an alternative approach using certain special biordered sets. As an application of our groupoid
+approach, in this section we provide an alternative construction. Although we of course arrive at
+the same semigroup (up to isomorphism), we believe our approach is a little more transparent.
+See Definition 8.19 and Theorems 8.20 and 8.21, and also Remark 8.22.
+For the rest of this section we fix a projection algebra P, as in Definition 3.1, and all of
+the data that comes with it, i.e. the unary operations θp, the relations ≤, ≤F and F, and so
+on. We also write P = P(P) and C = C (P) for the path category and chain groupoid of P;
+cf. Definitions 3.17 and 3.39.
+Recall that for p ∈ P, we have the down-set
+p↓ = {q ∈ P : q ≤ p}.
+If s, t ∈ p↓, then by Lemma 3.9 we have sθt = sθtθp ≤ p. This shows that p↓ is closed under each
+operation θt (t ∈ p↓), and is hence a projection algebra in its own right. (This is not to say it is
+a subalgebra of P, as it might not be closed under some θq for q ∈ P \ p↓. On the other hand,
+any down-set p↓ is a ⋄-subalgebra of P; cf. Remark 3.2.)
+Definition 8.5. For p, q ∈ P, a P-isomorphism p → q is a projection algebra isomorphism p↓ → q↓,
+i.e. a bijection α : p↓ → q↓ satisfying
+(sθt)α = (sα)θtα
+for all s, t ∈ p↓.
+(8.6)
+We write M(p, q) for the set of all such P-isomorphisms p → q, and we set
+M = M(P) =
+�
+p,q∈P
+M(p, q).
+It is easy to see that M is a groupoid, under ordinary function composition and inversion, and
+with objects/identities vM = {idp↓ : p ∈ P}. As usual we identify vM ≡ P, viz. p ≡ idp↓. We
+call M the Munn groupoid of P, in honour of the early work of Douglas Munn on fundamental
+inverse semigroups [109].
+Munn’s semigroups were constructed using order-isomorphisms of
+principal ideals/down-sets of semilattices, and this approach has been highly influential in studies
+of inverse semigroups and their various generalisations.
+We begin with the following simple lemma.
+85
+
+Lemma 8.7. Every α ∈ M is order-preserving, in the sense that
+s ≤ t
+⇒
+sα ≤ tα
+for all s, t ≤ d(α).
+Proof. We have s ≤ t ⇒ s = sθt ⇒ sα = (sθt)α = (sα)θtα ⇒ sα ≤ tα.
+For α ∈ M, and for p ≤ d(α), we have p↓ ⊆ d(α)↓ = dom(α), so we can define the restriction
+p⇃α = α|p↓.
+Here as usual we write f|A for the restriction of a function f to a subset A of its domain.
+Lemma 8.8. If α ∈ M, and if p ≤ d(α), then p⇃α ∈ M(p, pα). Consequently, p↓α = (pα)↓.
+Proof. Of course it suffices to prove the first claim. By definition we have p⇃α = α|p↓. For any
+t ∈ p↓, we have t ≤ p, and so tα ≤ pα by Lemma 8.7, and this says that tα ∈ (pα)↓. Thus, p⇃α
+maps p↓ into (pα)↓. Since α is injective and satisfies (8.6), so too does p⇃α. Finally, p⇃α is
+surjective onto (pα)↓, since for any t ≤ pα we have t = (tα−1)α, with tα−1 ≤ (pα)α−1 = p since
+α−1 ∈ M is order-preserving by Lemma 8.7.
+Lemma 8.9. For any projection algebra P, M = M(P) is an ordered groupoid.
+Proof. Again the proof is by an application of Lemma 2.3. We have the usual order ≤ on
+vM = P (cf. (3.5)), and we have already defined the restrictions p⇃α = α|p↓. It is then routine
+to check that properties (O1)′–(O5)′ all hold.
+As usual, the right-handed restrictions are defined by
+α⇂q = (⇃qα−1)−1 = (α−1|q↓)−1
+for α ∈ M and q ≤ r(α).
+Next we wish to verify that M = M(P) is a projection groupoid. To do so, we need to understand
+the ϑ and Θ maps. It follows from Lemma 8.8 that for α ∈ M, the map
+ϑα : d(α)↓ → r(α)↓
+is given by
+pϑα = r(p⇃α) = pα
+for all p ≤ d(α).
+In other words, we have ϑα = α (!) for all α ∈ M, so also Θα = θd(α)ϑα = θd(α)α. To summarise:
+ϑα = α
+and
+Θα = θd(α)α
+for all �� ∈ M.
+(8.10)
+Lemma 8.11. For any projection algebra P, M = M(P) is a projection groupoid.
+Proof. By Lemma 8.9 we just need to verify (G1).
+For this, it is essentially trivial to see
+that (G1d) holds, as ϑα = α is a projection algebra morphism by definition, for all α ∈ M.
+Consider a pair p, q ∈ P with p F q. Since the operation θq maps into q↓, so too does the
+restriction
+γpq = θq|p↓.
+Lemma 8.12. For any p, q ∈ P with p F q, we have γpq ∈ M(p, q), and γ−1
+pq = γqp.
+Proof. This can be proved directly, but it also follows by combining results of earlier chapters.
+Recall from Proposition 7.4 that (C , idC ) is a chained projection groupoid, where C = C (P) is
+the chain groupoid of P. Applying Lemma 4.14(ii) to this groupoid, and remembering that the
+evaluation map is ε = idC , we have
+ϑ[p,q] = ϑidC [p,q] = θq|p↓ = γpq.
+It follows from property (G1d) that γpq = ϑ[p,q] is a projection algebra isomorphism p↓ → q↓,
+i.e. that γpq ∈ M(p, q). Combining the above with Lemma 2.12, it follows that
+γqp = ϑ[q,p] = ϑ[p,q]−1 = ϑ−1
+[p,q] = γ−1
+pq .
+86
+
+To show that M is a chained projection groupoid, we need to define an evaluation map
+ε = ε(P) : C → M. As usual, it is convenient to first define a functor
+π = π(P) : P → M : (p1, . . . , pk) �→ γp1p2 · · · γpk−1pk.
+By convention, when k = 1, we interpret this last expression simply as idp↓
+1 ≡ p1. This means
+that π(p) = idp↓ ≡ p for all p ∈ P, i.e. that π is a v-functor. The next result concerns the
+congruence ≈ = Ω♯ on P from Definition 3.36.
+Lemma 8.13. We have ≈ ⊆ ker(π).
+Proof. We need to check that π(s) = π(t) for each pair (s, t) ∈ Ω. As usual this is clear when
+the pair has one of the forms (Ω1) or (Ω2). So suppose instead that
+s = λ(e, p, f) = (e, eθp, f)
+and
+t = ρ(e, p, f) = (e, fθp, f)
+for some projection p ∈ P, and some p-linked pair (e, f), as in Definition 3.31. Then
+π(s) = γe,eθpγeθp,f
+and
+π(t) = γe,fθpγfθp,f.
+Both π(s) and π(t) map e↓ → f↓, and we must show that these maps are equal, i.e. that
+qπ(s) = qπ(t) for any q ≤ e. By definition of the γuv, and keeping q = qθe in mind (as q ≤ e),
+we calculate
+qπ(s) = qθeθpθf =4 (qθe)θpθeθpθf =5 qθeθpθf = qθpθf
+We also have qπ(t) = qθfθpθf =4 qθpθfθpθf =5 qθpθf, so the proof is complete.
+Definition 8.14. Since C = P/≈, it follows from Lemma 8.13 that we have a well-defined
+functor
+ε = ε(P) : C → M
+given by
+ε[p] = π(p)
+for p ∈ P.
+That is, ε[p1, . . . , pk] = γp1p2 · · · γpk−1pk whenever p1 F · · · F pk (and ε[p] = p ≡ idp↓ for p ∈ P).
+The proof of the next result makes use of the basic fact that for bijections f : A → B and
+g : B → C, and for X ⊆ A, we have
+(f ◦ g)|X = f|X ◦ g|Y
+where Y = Xf.
+(8.15)
+(This is essentially (O5)′ in the category of bijections.)
+Lemma 8.16. The functor ε = ε(P) : C → M is an evaluation map.
+Proof. As with Lemma 5.17, the proof boils down to showing that
+π(q⇃p) = q⇃π(p)
+for all p ∈ P and q ≤ d(p).
+We prove this by induction on k, the length of the path p = (p1, . . . , pk). When k = 1, both sides
+evaluate to q ≡ idq↓. We now assume that k ≥ 2, and we write q⇃p = (q1, . . . , qk) as in (3.19).
+Also write p′ = (p1, . . . , pk−1), noting that q⇃p′ = (q1, . . . , qk−1). By induction we have
+π(q⇃p) = π(q1, . . . , qk) = γq1q2 · · · γqk−2qk−1γqk−1qk = π(q⇃p′) ◦ γqk−1qk = q⇃π(p′) ◦ γqk−1qk.
+On the other hand, using (8.15), and writing Y = q↓γp1p2 · · · γpk−2pk−1, we have
+q⇃π(p) = (γp1p2 · · · γpk−1pk)|q↓ = (γp1p2 · · · γpk−2pk−1)|q↓ ◦ γpk−1pk|Y = q⇃π(p′) ◦ γpk−1pk|Y .
+87
+
+Examining the last two conclusions, it remains to show that γpk−1pk|Y = γqk−1qk. By Lemma 8.8
+and (3.19), we have
+Y = q↓(γp1p2 · · · γpk−2pk−1) = (qγp1p2 · · · γpk−2pk−1)↓ = (qθp2 · · · θpk−1)↓ = q↓
+k−1,
+and so
+γpk−1pk|Y = (θpk|p↓
+k−1)|q↓
+k−1 = θpk|q↓
+k−1.
+Since γqk−1qk = θqk|q↓
+k−1, it remains to show that tθpk = tθqk for all t ∈ q↓
+k−1. For this we use
+t = tθqk−1 (as t ≤ qk−1), qk = qk−1θpk (by (3.19)), and the projection algebra axioms to calculate
+tθqk = (tθqk−1)θqk−1θpk =4 tθqk−1θpkθqk−1θpk =5 tθqk−1θpk = tθpk.
+Note in particular that ε[p, q] = γpq for p, q ∈ P with p F q.
+Proposition 8.17. If P is a projection algebra, then (M, ε) is a chained projection groupoid.
+Proof. By Lemmas 8.11 and 8.16, it remains to verify (G2). To do so, let (e, f) be a β-linked
+pair, where β ∈ M(q, r). Let the ei, fi be as in (4.19). We must show that λ = ρ, where
+λ = λ(e, β, f) = γee1 ◦ e1⇃β ◦ γf1f
+and
+ρ = ρ(e, β, f) = γee2 ◦ e2⇃β ◦ γf2f.
+Keeping γuv = θv|u↓ in mind, this is equivalent to showing that
+tθe1βθf = tθe2βθf
+for all t ≤ e.
+(8.18)
+Starting with the left-hand side, we use t = tθe (as t ≤ e), e1 = eθq (by (4.19)), and the projection
+algebra axioms to calculate
+tθe1βθf = tθeθeθqβθf =4 tθeθqθeθqβθf =5 tθeθqβθf = tθqβθf.
+On the other hand we have
+tθe2βθf = tθfΘβ−1βθf
+by (4.19)
+= tΘβθfΘβ−1βθf
+by (G1b)
+= tθqβθfθrβ−1βθf
+by (8.10)
+= tθqβθfθrθf
+= tθqβθf
+by Lemma 3.12(iii), as f ≤F r (cf. Lemma 4.18).
+Examining the previous two conclusions, we have completed the proof of (8.18), and hence of
+the proposition.
+Now that we know (M, ε) is a chained projection groupoid for any projection algebra P, we
+can apply the functor S to obtain a regular ∗-semigroup (cf. Theorems 4.36 and 6.7):
+Definition 8.19. For a projection algebra P, we define the regular ∗-semigroup
+MP = S(M, ε),
+where M = M(P) and ε = ε(P) are as in Definitions 8.5 and 8.14. Explicitly:
+• The elements of MP are the P-isomorphisms p↓ → q↓ (p, q ∈ P), as in Definition 8.5.
+• The product • in MP is given, for α, β ∈ MP with r(α) = p and d(β) = q, by
+α • β = α⇂p′ ◦ γp′q′ ◦ q′⇃β = (αθq′β)|(p′α−1)↓
+where
+p′ = qθp
+and
+q′ = pθq.
+88
+
+• The involution in MP is given by ordinary inversion of bijections, α∗ = α−1.
+• The projections of MP are the identity maps idp↓ ≡ p, for p ∈ P, and consequently
+P(MP ) ≡ P.
+• The idempotents of MP are the maps p • q = γpq, for (p, q) ∈ F.
+We call MP the Munn semigroup of P.
+An ostensibly weaker version of the next result was proved in [70, Theorem 5], albeit with a
+different formulation of MP ; cf. Remark 8.25. The statement, and the following one, refers to
+the canonical homomorphisms of Definition 7.9.
+Theorem 8.20. If S is a regular ∗-semigroup with projection algebra P(S) = P, then
+(i) there is a canonical ∗-semigroup homomorphism
+φS : S → MP
+given by
+aφS = ϑa
+for all a ∈ S,
+(ii) ker(φS) = µS is the maximum idempotent-separating congruence of S, and consequently S
+is fundamental if and only if φS is injective,
+(iii) im(φS) ∼= S/µS is the fundamental image of S.
+Proof. Throughout the proof we write M = M(P) and ε = ε(P), as in Definitions 8.5 and 8.14.
+We also write G(S) = (G, ε′).
+(i). By (G1d), we have ϑa ∈ M(p, q) for any a ∈ G(p, q). It follows that we have a well-defined
+mapping
+φ : G → M : a �→ ϑa.
+Our first goal is to show that φ is a chained projection functor (G, ε′) → (M, ε); cf. Definition 6.1.
+It follows quickly from Lemma 2.14 that φ is an ordered v-functor. Thus, (F1) holds trivially. It
+also follows from vφ = idP that the induced map Φ : C → C at the top of the diagram in (F2)
+is also an identity map. Thus, the diagram becomes:
+C
+C
+G
+M,
+idC
+ε′
+ε
+φ
+and we must show that this commutes, i.e. that
+(ε′(c))φ = ε(c)
+for all c ∈ C .
+This is clear if c = [p] ≡ p for some p ∈ P, so suppose instead that c = [p1, . . . , pk] with k ≥ 2.
+Then using the definitions, and the fact that each of the maps in the diagram is a functor, we
+calculate
+(ε′(c))φ = (ε′[p1, p2] ◦ · · · ◦ ε′[pk−1, pk])φ
+= (ε′[p1, p2])φ ◦ · · · ◦ (ε′[pk−1, pk])φ
+= ϑε′[p1,p2] ◦ · · · ◦ ϑε′[pk−1,pk]
+= θp2|p↓
+1 ◦ · · · ◦ θpk|p↓
+k−1
+by Lemma 4.14(ii)
+= γp1p2 ◦ · · · ◦ γpk−1pk
+= ε[p1, p2] ◦ · · · ◦ ε[pk−1, pk] = ε(c),
+89
+
+as required.
+We now know that φ is a chained projection functor G(S) = (G, ε′) → (M, ε). By Lemma 6.3
+(and Theorem 6.7), φ is also a ∗-semigroup homomorphism S = S(G(S)) → S(M, ε) = MP , and
+of course φ = φS is the map in the statement of the theorem. It is canonical because φ is a
+v-functor.
+(ii). This follows immediately from Proposition 8.3 and the definition of φS.
+(iii). This follows from the fundamental homomorphism theorem.
+We can now quickly deduce the next result, which is essentially [72, Theorem 3.2]. It says that
+not only is MP fundamental, but it is in fact the maximum fundamental regular ∗-semigroup
+with projection algebra P.
+Theorem 8.21. If P is a projection algebra, then
+(i) MP is a fundamental regular ∗-semigroup with projection algebra P,
+(ii) any fundamental regular ∗-semigroup with projection algebra P embeds canonically in MP .
+Proof. (i). By (8.10), the homomorphism φMP : MP → MP : α �→ ϑα from Theorem 8.20(i) is
+simply the identity map, and is therefore of course injective. Consequently, MP is fundamental
+by Theorem 8.20(ii). We have already observed that P(MP ) is (isomorphic to) P.
+(ii). This follows immediately from Theorem 8.20(ii).
+Remark 8.22. The above construction of MP using P-isomorphisms is closest in spirit to the
+work of Yamada [138] and Jones [73], although Yamada’s basic set-up was quite different; in
+place of projection algebras, he used ‘P-sets in fundamental regular warps’.
+An alternative construction of MP was given by Imaoka [70], and we briefly comment here
+on how to interpret this in our set-up. Let TP denote the full transformation semigroup over P,
+i.e. the semigroup of all (totally-defined) functions P → P, under composition. For any regular
+∗-semigroup S with P(S) = P, Lemma 4.29 guarantees the existence of an (ordinary semigroup)
+homomorphism
+S → TP : a �→ Θa.
+(Recall that the • operation in S(G(S)) = S is simply the original product in S.) Since the
+involution of S is an antihomomorphism, the map a �→ Θa∗ is an antihomomorphism S → TP ,
+but an (ordinary) homomorphism S → T op
+P . The latter denotes the opposite semigroup to TP ;
+the product ⋆ in T op
+P
+is given by α ⋆ β = βα. Thus, we have a homomorphism into the direct
+product
+ξS : S → TP × T op
+P
+given by
+aξS = (Θa, Θa∗)
+for a ∈ S.
+By Proposition 8.3, ker(ξS) = µS is the maximum idempotent-separating congruence of S, and
+im(ξS) ∼= S/µS is the fundamental image of S. In particular, when S = MP , the image of ξMP
+is isomorphic to MP . Keeping (8.10) in mind, this copy of MP , which we will denote by MP ,
+is precisely the subsemigroup
+MP = im(ξMP ) = {α : α ∈ MP } ≤ TP × T op
+P ,
+where we write α = (θpα, θqα−1) for α ∈ M(p, q). The product in MP is simply composition
+(and reverse composition) of transformations. While TP and T op
+P
+are not regular ∗-semigroups
+for |P| ≥ 2 (as their D-classes are not square), of course MP is, and its involution is given
+by α∗ = α−1. Returning to the general case, the image of ξS (where S is an arbitrary regular
+∗-semigroup with P(S) = P) is contained in MP , and ξS is an embedding S → MP if and only
+if S is fundamental.
+90
+
+Another, very different, way to construct MP can be found in [113, Section 3], wherein the
+projection algebra P is characterised as a specialised poset with a connection to a certain dual
+poset P ◦ (in the sense of Grillet [60,61]), and MP is realised as a subdirect product, similar to
+Imaoka’s construction.
+We believe the following problem is of considerable interest:
+Problem 8.23. Given a projection algebra P, describe the maximum fundamental regular ∗-
+semigroup MP .
+In particular, one could attempt to do this in the case that P = P(S) arises from some
+natural regular ∗-semigroup S, or for some family of such semigroups. For example, the case
+that S is a finite diagram monoid (including partition monoids) is the subject of an ongoing work
+by the current authors [36].
+8.3
+Idempotent-generated fundamental regular ∗-semigroups
+Finally, we combine the ideas of this chapter and the previous one. The next result concerns
+the chain semigroup CP , as in Definition 7.8, and the proof utilises the map φCP : CP → MP
+from Theorem 8.20. The fundamental image of CP is the quotient CP /µ, where µ = µCP is the
+maximum idempotent-separating congruence. This quotient is of course idempotent-generated,
+as CP is. For the following statement, recall that E(S) denotes the idempotent-generated sub-
+semigroup of the semigroup S. To the best of our knowledge, no general results exist in the
+literature concerning idempotent-generated fundamental regular semigroups, though they were
+used tangentially in the proof of [110, Theorem 7.2] in the case of so-called solid biordered sets.
+Theorem 8.24. Let P be a projection algebra, and let µ = µCP be the maximum idempotent-
+separating congruence on the chain semigroup CP .
+(i) Up to isomorphism, there is exactly one idempotent-generated fundamental regular ∗-semigroup
+with projection algebra P, namely CP /µ.
+(ii) For any fundamental regular ∗-semigroup S with projection algebra P = P(S), we have
+E(S) ∼= CP /µ.
+Proof. We prove both parts together. Suppose S is an arbitrary fundamental regular ∗-semigroup
+with projection algebra P = P(S). By Proposition 7.10 and Theorem 8.20, we have the two
+∗-semigroup homomorphisms
+CP
+S
+MP .
+ε
+φS
+Here, ε is the evaluation map C → G(S), and φS is injective by Theorem 8.20(ii). Since all
+three of the above semigroups have projection algebra P, and since both of the above maps is
+canonical, it follows that the composition
+εφS : CP → MP
+is also canonical.
+But CP is projection-generated, so there can be at most one canonical ∗-
+semigroup homomorphism CP → MP . It follows that in fact εφS = φCP . But then
+ker(ε) = ker(εφS)
+as φS is injective
+= ker(φCP )
+as εφS = φCP
+= µCP = µ
+by Theorem 8.20(ii).
+91
+
+Combining this with Proposition 4.39(iii) and the fundamental homomorphism theorem, it fol-
+lows that
+E(S) = im(ε) ∼= CP / ker(ε) = CP /µ.
+This of course gives (ii). Part (i) also follows, since if S happens to be idempotent-generated
+(and fundamental, with P(S) = P), then S = E(S) ∼= CP /µ.
+Remark 8.25. Theorem 8.24 shows that (up to isomorphism) there is a unique idempotent-
+generated fundamental regular ∗-semigroup with a given projection algebra P. The theorem also
+gives a number of ways to get hold of such a semigroup, which for now we will denote by FP .
+First, we can construct FP as the quotient CP /µ, where µ = µCP . The elements of this quotient
+are µ-classes of P-chains. For two P-chains c = [p1, . . . , pk] and d = [q1, . . . , ql], we combine
+Proposition 8.3 with (7.1) and (7.2) to obtain
+(c, d) ∈ µ ⇔ θp1 · · · θpk = θq1 · · · θql
+and
+θpk · · · θp1 = θql · · · θq1
+⇔ θp1 · · · θpk = θq1 · · · θql
+and
+p1 = q1
+⇔ θp2 · · · θpk = θq2 · · · θql
+and
+p1 = q1.
+Alternatively, we could take FP to be the idempotent-generated subsemigroup E(MP ) of MP .
+Elements of E(MP ) are of the form
+idp↓
+1 • · · · • idp↓
+k
+for p1, . . . , pk ∈ P with p1 F · · · F pk.
+As in Definition 8.19, for such pi we have
+idp↓
+1 • idp↓
+2 • · · · • idp↓
+k = γp1p2γp2p3 · · · γpk−1pk = (θp2 · · · θpk)|p↓
+1,
+so E(MP ) consists of all such maps.
+As another possibility, we could take FP = E(MP ), where MP ≤ TP ×T op
+P
+is the isomorphic
+copy of MP discussed in Remark 8.22.
+Using the over-line notation from that remark, and
+using (4.2), the projections of MP are of the form
+p = (Θp, Θp∗) = (θp, θp)
+for p ∈ P.
+Keeping in mind that the operation in MP is (ordinary and reverse) composition, it follows that
+a general element of E(MP ) has the form
+(θp1 · · · θpk, θpk · · · θp1)
+for p1, . . . , pk ∈ P with p1 F · · · F pk.
+9
+Inverse semigroups and inductive groupoids
+We have noted on a number of occasions that any inverse semigroup is a regular ∗-semigroup
+(with a∗ = a−1). In this final chapter, we look at how the theory developed above simplifies
+in the case of inverse semigroups. In particular, we will see that our results allow us to deduce
+the celebrated Ehresmann–Schein–Nambooripad (ESN) Theorem, stated below as Theorem 9.1.
+This theorem was first explicitly formulated by Lawson in [87, Theorem 4.1.8], who named it as
+such in order to honour the contributions of the three stated mathematicians to the development
+of the result. We discussed this in some detail in Chapter 1, but see [87, Chapter 4] and [66,67]
+for more.
+In what follows, we write IS for the category of inverse semigroups. Morphisms in IS are sim-
+ply the semigroup homomorphisms. (Any semigroup homomorphism between inverse semigroups
+automatically respects the involutions.) We also write IG for the category of inductive groupoids,
+i.e. the ordered groupoids whose object set is a semilattice (under the order inherited from the
+containing groupoid). Morphisms in IG are the inductive functors, i.e. the ordered groupoid
+functors whose object maps are semilattice morphisms. The ESN Theorem is as follows:
+92
+
+Theorem 9.1. The category IS of inverse semigroups (with semigroup homomorphisms) is iso-
+morphic to the category IG of inductive groupoids (with inductive functors).
+We will give a proof of this theorem in Section 9.2, relying on our above results on regular
+∗-semigroups. But before we do, in Section 9.1 we show how we are somewhat-inevitably led to
+the ESN Theorem by considering the simplifications that arise when we specialise our general
+theory to inverse semigroups.
+9.1
+The chained projection groupoid associated to an inverse semigroup
+In Theorem 6.7 we showed that the functors
+G : RSS → CPG
+and
+S : CPG → RSS
+are mutually inverse isomorphisms between
+• the category RSS of regular ∗-semigroups, with ∗-semigroup homomorphisms, and
+• the category CPG of chained projection groupoids, with chained projection functors.
+It follows that for any subcategory C of RSS, the functor G restricts to an isomorphism from C
+onto its image G(C) in CPG:
+RSS
+C
+CPG
+G(C)
+G
+S
+In particular, the category IS of inverse semigroups is isomorphic to its image G(IS). To under-
+stand this image, fix an inverse semigroup S, and let (G, ε) = G(S) be the chained projection
+groupoid associated to S, as in Definitions 5.4 and 5.16. As usual we write
+E = E(S) = {e ∈ S : e2 = e}
+and
+P = P(S) = {p ∈ S : p2 = p = p−1}
+for the semilattice of idempotents, and the projection algebra of S, respectively. Since each idem-
+potent is self-inverse, it follows that in fact P = E. As there will be a number of orders in play, we
+will (temporarily) write ≤′ for the natural partial order on E, defined by e ≤′ f ⇔ e = ef(= fe).
+Note then that the (order-theoretic) meet of two idempotents is their product (in S), e∧f = ef,
+and so e ≤′ f ⇔ e = e ∧ f.
+As in (5.1), and remembering that idempotents commute, the θ maps on E(= P) are given
+by:
+eθf = fef = ef = e ∧ f = fθe
+for all e, f ∈ E.
+(9.2)
+It follows quickly from this that the ≤ and ≤F relations, defined on E(= P) in (3.5) and (3.10),
+coincide with ≤′ defined above. In particular, ≤F is a partial order, and so F = ≤F ∩≥F = ∆E
+is the trivial relation on E. It follows that the only E-paths are of the form (e, e, . . . , e), and each
+such path is ≈-equivalent to (e) ≡ e. This all shows that the chain groupoid C = C (E) is trivial,
+meaning that it consists entirely of its identity morphisms. That is, we have C = vC = E, and
+e, f ∈ E are composable in C precisely when e = f. It follows that the evaluation map ε : C → G
+(being a v-functor) is just the inclusion ι : E �→ G(= S). This means that all the information
+contained in the pair (G, ε) = (G, ι) is in fact contained in the groupoid G = G(S) alone.
+Roughly speaking, the next result shows that the above situation is reversible.
+93
+
+Proposition 9.3. Let (G, ε) be a chained projection groupoid, and let P = vG and C = C (P).
+Then (G, ε) = G(S) for some inverse semigroup S if and only if C = P is trivial (in which case ε
+is the inclusion ε = ι : P �→ G).
+Proof. We have already proved the forwards implication. Conversely, assume C = P is trivial,
+and (hence) ε is the inclusion ε = ι : P �→ G. Let S = S(G, ι), so that S is a regular ∗-semigroup
+with projection algebra P(S) = P, and (G, ι) = G(S); cf. Theorem 6.7. We must show that S is
+inverse, and we can do this by showing that idempotents commute. As usual, we write E = E(S)
+for the set of idempotents of S. By Proposition 4.39(ii), E is contained in im(ε) = im(ι), and
+by assumption the latter is equal to P. Since P ⊆ E for any regular ∗-semigroup, it follows
+that E = P. But also E = P 2 by Lemma 2.23(ii), so it follows that P = P 2, i.e. that P is
+a subsemigroup of S. Now let e, f ∈ E = P. Since e, f and ef are all projections, we have
+ef = (ef)∗ = f∗e∗ = fe, as required.
+Remark 9.4. Given an inverse semigroup S, the groupoid G = G(S) is inductive, as the object
+set vG = E = E(S) is a semilattice. At the outset, one might expect there to also be a version
+of Proposition 9.3 stating that a chained projection groupoid (G, ε) corresponds to an inverse
+semigroup if and only if G is inductive. However, it follows from Example 5.21 that this is not the
+case. Indeed, there we considered two regular ∗-semigroups S1 and S2, with S2 inverse and S1
+non-inverse, and with the same groupoids G(S1) = G(S2). Since S2 is inverse, this groupoid
+is inductive. It follows that the non-inverse regular ∗-semigroup S1 gives rise to an inductiive
+groupoid G(S1).
+The next result is essentially a reformulation of Proposition 9.3, but it seems to be worth
+recording.
+Proposition 9.5. Let S be a regular ∗-semigroup, with projection algebra P = P(S). Then S is
+inverse if and only if F = ∆P .
+Proof. Clearly F = ∆P is equivalent to C = P, and the result then follows from Proposition 9.3
+(and Theorem 6.7).
+Now that we have some understanding of the image G(IS) of IS in CPG (cf. Proposition 9.3),
+we are a step closer to proving Theorem 9.1.
+One might now go on to show that G(IS) is
+isomorphic to IG, the category of inductive groupoids. However, there are still some subtle
+points remaining to be dealt with, some so subtle as to be almost invisible at this point. Rather
+than continue on this path, we take a more direct/canonical route in the next section, though
+we will make use of Proposition 9.3 on a number of occasions.
+9.2
+The Ehresmann–Schein–Nambooripad Theorem
+In Proposition 9.3 we characterised the chained projection groupoids (G, ε) corresponding to
+inverse semigroups; essentially the proposition shows that these are precisely those in which the
+evaluation map ε carries no information whatsoever. Consequently, if we wish to prove the ESN
+Theorem (stated above as Theorem 9.1) we may as well dispense with evaluation maps altogether
+(when dealing with inverse semigroups), and consider the groupoid construction S �→ G(S) as
+a functor IS → IG, and prove that this is a category isomorphism.
+This is of course the
+standard/canonical approach to the ESN Theorem, but the details required for the proof will be
+taken from our general theory developed in earlier chapters. Following this canonical approach,
+we proceed by defining two functors
+G : IS → IG
+and
+S : IG → IS,
+and showing that they are mutually inverse isomorphisms.
+94
+
+First, for an inverse semigroup S, we take G(S) = G = G(S) to be the ordered groupoid
+constructed in Definition 5.4. As in Section 9.1, we have E = P (where as usual E = E(S) and
+P = P(S)). Moreover, following Definition 5.4, for e, f ∈ E we have
+e ≤ f in G
+⇔
+e = ef in S
+⇔
+e ≤ f in E.
+In particular, E = vG is a semilattice (under the order inherited from G), and so G(S) = G is
+indeed an inductive groupoid. Next, it is easy to see that any morphism φ : S → S′ in IS is an
+inductive functor G(S) → G(S′). Indeed, φ is an ordered functor by Lemma 6.2, and the object
+map vφ is a semilattice morphism since for any e, f ∈ vG(S) = E(S) we have
+(e ∧ f)φ = (ef)φ = (eφ)(fφ) = eφ ∧ fφ.
+So we can simply take G(φ) = φ for any morphism φ from IS.
+To define the functor S, consider an inductive groupoid G, with object semilattice E = vG.
+The semigroup S = S(G) will have the same underlying set as G. To define the product on S,
+which we will denote by ⋆, consider two morphisms a, b ∈ G(= S), and write e = r(a) and
+f = d(b). Since E is a semilattice, the meet g = e ∧ f exists, and since g ≤ e, f we can define
+a ⋆ b = a⇂g ◦ g⇃b. At this point it is not at all clear that S = S(G) = (G, ⋆) is a semigroup, let
+alone inverse, but we will deal with this shortly. For any morphism φ : G → G′ in IG, we again
+define S(φ) = φ.
+Lemma 9.6. If G is an inductive groupoid, with object semilattice vG = E, then
+(i) E is a projection algebra with respect to the maps
+θe : E → E
+given by
+fθe = e ∧ f
+for e, f ∈ E,
+(ii) G is a projection groupoid.
+Proof. (i). It is a routine matter to check that axioms (P1)–(P5) hold.
+(ii). It remains to show that condition (G1) from Definition 4.8 holds, and for this we verify (G1d).
+To do so, fix some a ∈ G, and write p = d(a) and q = r(a). We must show that ϑa : p↓ → q↓ is
+a projection algebra morphism, i.e. that
+(eθf)ϑa = (eϑa)θfϑa
+for all e, f ≤ p.
+By the definition of the θ maps, this amounts to showing that
+(e ∧ f)ϑa = eϑa ∧ fϑa
+for all e, f ≤ p,
+i.e. that ϑa is a semilattice morphism. But this follows quickly from basic order-theoretic facts:
+• p↓ and q↓ are both meet semilattices, as order ideals of the semilattice E, and
+• ϑa : p↓ → q↓ and its inverse ϑ−1
+a
+= ϑa−1 : q↓ → p↓ are both order-preserving, by Lemma 2.15
+(cf. Lemma 2.12).
+It follows that ϑa preserves meets.
+95
+
+Lemma 9.7. If G is an inductive groupoid, with object semilattice vG = E, then
+(i) C (E) = E is trivial (where the projection algebra structure on E is as in Lemma 9.6),
+(ii) (G, ι) is a chained projection groupoid, where ι : E �→ G is the inclusion,
+(iii) S(G) = S(G, ι) is an inverse semigroup.
+Proof. (i). For any e, f ∈ E we have fθe = e ∧ f = f ∧ e = eθf. We then obtain F = ∆E, and
+hence C (E) = E, in exactly the same way as in the discussion following (9.2).
+(ii). By Lemma 9.6, it remains to check that (G2) holds, and this is essentially trivial. Indeed,
+consider some b-linked pair (e, f), where b ∈ G, and let e1, e2, f1, f2 ∈ E be as in (4.19). Since
+F = ∆E, it follows from Lemma 4.18(iii) that in fact e1 = e2 = e and f1 = f2 = f. But then
+λ(e, b, f) = ι[e, e] ◦ e⇃b ◦ ι[f, f] = e ◦ e⇃b ◦ f = e⇃b
+and similarly
+ρ(e, b, f) = e⇃b.
+(iii).
+The underlying sets of S(G) and S(G, ι) are both G.
+To check that the operations ⋆
+and • coincide, let a, b ∈ G. Write e = r(a) and f = d(b), and also set g = e ∧ f. Following
+Definition 4.24 (applied to the chained projection groupoid (G, ι)), we have
+a • b = a⇂e′ ◦ ι[e′, f′] ◦ f′⇃b
+where
+e′ = fθe
+and
+f′ = eθf.
+But e′ = fθe = e ∧ f = g, and similarly f′ = g, so in fact
+a • b = a⇂g ◦ ι[g, g] ◦ g⇃b = a⇂g ◦ g ◦ g⇃b = a⇂g ◦ g⇃b = a ⋆ b.
+It remains to check that the regular ∗-semigroup S = S(G, ι) is inverse, and this follows from
+Proposition 9.3 and part (i) of the current lemma, as (G, ι) = G(S).
+We call (G, ι), as in Lemma 9.7(ii), the trivial chained projective groupoid associated to the
+inductive groupoid G.
+Lemma 9.8. If G1 and G2 are inductive groupoids, then any inductive functor G1 → G2 is a
+semigroup homomorphism S(G1) → S(G2).
+Proof. Let φ : G1 → G2 be an inductive functor. By Lemmas 6.3 and 9.7(iii), it suffices to show
+that φ is a chained projection functor (G1, ι1) → (G2, ι2), where these are the trivial chained
+projective groupoids associated to G1 and G2.
+Following Definition 6.1, it remains to verify
+conditions (F1) and (F2).
+(F1). This follows from the fact that vφ is a semilattice morphism (as φ is inductive), and keeping
+the rule eθf = e ∧ f in mind.
+(F2). Writing Ei = vGi (i = 1, 2), each Ci = Ei is trivial, so the induced map Φ at the top of
+the diagram in Definition 6.1 is just the object map Φ = vφ. The diagram then becomes:
+E1
+E2
+G1
+G2,
+vφ
+ι1
+ι2
+φ
+and this obviously commutes.
+96
+
+It follows immediately from Lemmas 9.7(iii) and 9.8 that S is indeed a functor IG → IS. We
+can now tie together the loose ends:
+Proof of Theorem 9.1. It remains to show that S and G are mutually inverse, i.e. that
+(i) S(G(S)) = S for any inverse semigroup S, and
+(ii) G(S(G)) = G for any inductive groupoid G.
+Beginning with (i), fix an inverse semigroup S, and let G = G(S) = G(S). By Proposition 9.3,
+G(S) = (G, ι) is the trivial chained projection groupoid associated to G. Combining this with
+Lemmas 9.7(iii) and 6.6, it follows that
+S(G(S)) = S(G) = S(G, ι) = S(G(S)) = S.
+For (ii), fix an inductive groupoid G, and write S = S(G). By Lemma 9.7(iii) we have S = S(G, ι),
+where (G, ι) is the trivial chained projection groupoid associated to G. Combining this with
+Lemma 6.4 (and Definitions 5.4 and 5.16), it follows that
+(G, ι) = G(S(G, ι)) = G(S) = (G(S), ε(S)),
+and so G = G(S) = G(S) = G(S(G)).
+We have now seen that the Ehresmann–Schein–Nambooripad Theorem (Theorem 9.1) follows
+from our Theorem 6.7. Alternatively, one could also obtain a direct (and quite transparent) proof
+of the ESN Theorem by specialising our proof of Theorem 6.7 to the inverse case. We will not
+give the full outline of this, but will comment briefly on the most involved step, which is to show
+that the product • defined on the groupoid G is associative. In the inverse/inductive case, this
+product is defined by
+a • b = a⇂e ◦ e⇃b
+for a, b ∈ G, where e = r(a) ∧ d(b).
+It follows that for a, b, c ∈ G, we have
+(a • b) • c = a⇂f ◦ b⇂g ◦ c⇂h
+and
+a • (b • c) = a⇂f′ ◦ b⇂g′ ◦ c⇂h′
+for some f, g, h, f′, g′, h′ ∈ E = vG. We can then show that (a • b) • c and a • (b • c) are equal
+by showing that they have equal domains and equal ranges. This follows from (4.30) and (4.31),
+which were proved quite early in Section 4.3.
+9.3
+Fundamental inverse semigroups
+So far we have examined the way that the general theory developed in Chapters 3–6 simplifies
+in the case of inverse semigroups. Specifically, we were able to deduce the Ehresmann–Schein–
+Nambooripad Theorem (stated in Theorem 9.1) from our results. It is then of course natural to
+follow the same program for the results of Chapters 7 and 8.
+The results of Chapter 7 become trivial/vacuous in the context of inverse semigroups, as
+idempotent-generated inverse semigroups are simply inverse semigroups consisting entirely of
+idempotents, i.e. semilattices. It is therefore no surprise that the free (projection-generated)
+regular ∗-semigroup over a semilattice is simply the semilattice itself. There is another obvious
+way to to think about this. Given a semilattice E, regarded as a projection algebra with θ
+maps as in (9.2), we have already seen that the chain groupoid C = C (E) is trivial, in the
+sense that C = vC = E, with e, f ∈ E being composable in C precisely when e = f. Recall
+from Definition 3.39 that the chain semigroup of E is the semigroup CE = S(C , idC ). Since idC
+97
+
+can also be seen as the inclusion idC = ι : E = C �→ C , it follows from Lemma 9.7(iii)
+that CE = S(C , idC ) = S(C , ι) = S(C ). This semigroup has underlying set C = E, and the
+product • = ⋆ is given by
+e • f = e ⋆ f = e⇂e∧f ◦ e∧f⇃f = (e ∧ f) ◦ (e ∧ f) = e ∧ f.
+In other words, CE is just E itself.
+On the other hand, the results of Chapter 8 do have interesting specialisations to inverse
+semigroups, as we now briefly discuss. Fix a semilattice E, which we again consider as a pro-
+jection algebra, with θ maps given in (9.2). As in the proof of Lemma 9.6(ii), the projection
+algebra morphisms e↓ → f↓ (for e, f ∈ E) are precisely the semilattice morphisms e↓ → f↓.
+Since the order and meet operations are inter-definable in E (viz. s ≤ t ⇔ s = s ∧ t), it follows
+that these morphisms are also precisely the order isomorphisms e↓ → f↓. It follows that the
+(inductive) groupoid M = M(E) from Definition 8.5 consists precisely of all such order isomor-
+phisms e↓ → f↓ (e, f ∈ E). The resulting semigroup ME = S(M) has underlying set M, and
+product • = ⋆ given as follows. For α, β ∈ ME with r(α) = e and d(β) = f, we have
+α • β = α ⋆ β = α⇂e∧f ◦ e∧f⇃β.
+This last expression is the composition in the groupoid M of the bijections α⇂e∧f and e∧f⇃β; the
+(set-theoretic) range of the former is equal to the (set-theoretic) domain of the latter, and this
+is precisely the order ideal e↓ ∩ f↓ = (e ∧ f)↓. However, we can also think of α and β as partial
+bijections of E, i.e. as elements of the symmetric inverse monoid IE, and the above product
+α • β = α ⋆ β is simply the product of α and β in IE, i.e. their composition as binary relations:
+α • β = α ⋆ β = {(s, t) ∈ E × E : (s, g) ∈ α and (g, t) ∈ β for some g ∈ E}.
+In this way, we can think of ME as an inverse subsemigroup of the symmetric inverse monoid IE.
+This is in fact the original approach taken by Munn in his seminal paper [109], where the
+semigroup ME (considered as a subsemigroup of IE) was denoted TE.
+References
+[1] S. Armstrong. Structure of concordant semigroups. J. Algebra, 118(1):205–260, 1988.
+[2] K. Auinger. Krohn-Rhodes complexity of Brauer type semigroups. Port. Math., 69(4):341–360, 2012.
+[3] K. Auinger. Pseudovarieties generated by Brauer type monoids. Forum Math., 26(1):1–24, 2014.
+[4] K. Auinger, Y. Chen, X. Hu, Y. Luo, and M. V. Volkov. The finite basis problem for Kauffman monoids.
+Algebra Universalis, 74(3-4):333–350, 2015.
+[5] S. Awodey. Category theory, volume 52 of Oxford Logic Guides. Oxford University Press, Oxford, second
+edition, 2010.
+[6] G. Benkart and T. Halverson. Motzkin algebras. European J. Combin., 36:473–502, 2014.
+[7] G. Benkart and T. Halverson. Partition algebras Pk(n) with 2k > n and the fundamental theorems of
+invariant theory for the symmetric group Sn. J. Lond. Math. Soc. (2), 99(1):194–224, 2019.
+[8] G. Birkhoff. Rings of sets. Duke Math. J., 3(3):443–454, 1937.
+[9] M. Borisavljević, K. Došen, and Z. Petrić. Kauffman monoids. J. Knot Theory Ramifications, 11(2):127–143,
+2002.
+[10] R. Brauer. On algebras which are connected with the semisimple continuous groups. Ann. of Math. (2),
+38(4):857–872, 1937.
+[11] M. Brittenham, S. W. Margolis, and J. Meakin. Subgroups of the free idempotent generated semigroups
+need not be free. J. Algebra, 321(10):3026–3042, 2009.
+[12] N. Brownlowe, N. S. Larsen, and N. Stammeier. On C∗-algebras associated to right LCM semigroups.
+Trans. Amer. Math. Soc., 369(1):31–68, 2017.
+[13] S. Burris and H. P. Sankappanavar. A course in universal algebra, volume 78 of Graduate Texts in Mathe-
+matics. Springer-Verlag, New York-Berlin, 1981.
+98
+
+[14] A. Buss, R. Exel, and R. Meyer. Reduced C∗-algebras of Fell bundles over inverse semigroups. Israel J.
+Math., 220(1):225–274, 2017.
+[15] D. M. Clark and B. A. Davey. Natural dualities for the working algebraist, volume 57 of Cambridge Studies
+in Advanced Mathematics. Cambridge University Press, Cambridge, 1998.
+[16] A. H. Clifford and G. B. Preston. The algebraic theory of semigroups. Vol. I. Mathematical Surveys, No.
+7. American Mathematical Society, Providence, R.I., 1961.
+[17] Y. Dandan, I. Dolinka, and V. Gould. A group-theoretical interpretation of the word problem for free
+idempotent generated semigroups. Adv. Math., 345:998–1041, 2019.
+[18] D. DeWolf and D. Pronk. The Ehresmann-Schein-Nambooripad theorem for inverse categories. Theory
+Appl. Categ., 33:Paper No. 27, 813–831, 2018.
+[19] I. Dolinka. Free idempotent generated semigroups: the word problem and structure via gain graphs. Israel
+J. Math., 245(1):347–387, 2021.
+[20] I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, J. Hyde, and N. Loughlin. Enumeration of
+idempotents in diagram semigroups and algebras. J. Combin. Theory Ser. A, 131:119–152, 2015.
+[21] I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, J. Hyde, N. Loughlin, and J. D. Mitchell.
+Enumeration of idempotents in planar diagram monoids. J. Algebra, 522:351–385, 2019.
+[22] I. Dolinka, J. East, and R. D. Gray. Motzkin monoids and partial Brauer monoids. J. Algebra, 471:251–298,
+2017.
+[23] I. Dolinka and R. D. Gray. Maximal subgroups of free idempotent generated semigroups over the full linear
+monoid. Trans. Amer. Math. Soc., 366(1):419–455, 2014.
+[24] I. Dolinka, R. D. Gray, and N. Ruškuc. On regularity and the word problem for free idempotent generated
+semigroups. Proc. Lond. Math. Soc. (3), 114(3):401–432, 2017.
+[25] I. Dolinka and N. Ruškuc. Every group is a maximal subgroup of the free idempotent generated semigroup
+over a band. Internat. J. Algebra Comput., 23(3):573–581, 2013.
+[26] M. P. Drazin. Regular semigroups with involution. In Symposium on Regular Semigroups, pages 29–46.
+Northern Illinois University, DeKalb, 1979.
+[27] D. Easdown. Biordered sets are biordered subsets of idempotents of semigroups. J. Austral. Math. Soc.
+Ser. A, 37(2):258–268, 1984.
+[28] D. Easdown. Biordered sets of bands. Semigroup Forum, 29(1-2):241–246, 1984.
+[29] D. Easdown. Biordered sets of eventually regular semigroups. Proc. London Math. Soc. (3), 49(3):483–503,
+1984.
+[30] D. Easdown.
+A new proof that regular biordered sets come from regular semigroups.
+Proc. Roy. Soc.
+Edinburgh Sect. A, 96(1-2):109–116, 1984.
+[31] D. Easdown. Biordered sets come from semigroups. J. Algebra, 96(2):581–591, 1985.
+[32] D. Easdown. Biordered sets of rings. In Monash Conference on Semigroup Theory (Melbourne, 1990), pages
+43–49. World Sci. Publ., River Edge, NJ, 1991.
+[33] J. East. Generators and relations for partition monoids and algebras. J. Algebra, 339:1–26, 2011.
+[34] J. East. On the singular part of the partition monoid. Internat. J. Algebra Comput., 21(1-2):147–178, 2011.
+[35] J. East. Infinite partition monoids. Internat. J. Algebra Comput., 24(4):429–460, 2014.
+[36] J. East and P. A. Azeef Muhammed. Fundamental diagram monoids. In preparation.
+[37] J. East, P. A. Azeef Muhammed, and N. Ruškuc. Free idempotent-generated regular ∗-semigroups. In
+preparation.
+[38] J. East and D. G. FitzGerald. The semigroup generated by the idempotents of a partition monoid. J.
+Algebra, 372:108–133, 2012.
+[39] J. East and R. D. Gray. Diagram monoids and Graham–Houghton graphs: Idempotents and generating
+sets of ideals. J. Combin. Theory Ser. A, 146:63–128, 2017.
+[40] J. East, J. D. Mitchell, N. Ruškuc, and M. Torpey. Congruence lattices of finite diagram monoids. Adv.
+Math., 333:931–1003, 2018.
+[41] J. East and N. Ruškuc. Classification of congruences of twisted partition monoids. Adv. Math., 395:Paper
+No. 108097, 65 pp., 2022.
+[42] J. East and N. Ruškuc. Congruences on infinite partition and partial Brauer monoids. Mosc. Math. J.,
+22(2):295–372, 2022.
+[43] J. East and N. Ruškuc. Properties of congruences of twisted partition monoids and their lattices. J. Lond.
+Math. Soc. (2), 106(1):311–357, 2022.
+99
+
+[44] J. East and N. Ruškuc. Congruence lattices of ideals in categories and (partial) semigroups. Mem. Amer.
+Math. Soc., to appear, arXiv:2001.01909 .
+[45] C. Ehresmann. Gattungen von lokalen Strukturen. Jber. Deutsch. Math.-Verein., 60(Abt. 1):49–77, 1957.
+[46] C. Ehresmann. Catégories inductives et pseudo-groupes. Ann. Inst. Fourier (Grenoble), 10:307–332, 1960.
+[47] C. Farthing, P. S. Muhly, and T. Yeend. Higher-rank graph C∗-algebras: an inverse semigroup and groupoid
+approach. Semigroup Forum, 71(2):159–187, 2005.
+[48] D. G. Fitz-Gerald. On inverses of products of idempotents in regular semigroups. J. Austral. Math. Soc.,
+13:335–337, 1972.
+[49] D. G. FitzGerald. Factorizable inverse monoids. Semigroup Forum, 80(3):484–509, 2010.
+[50] D. G. FitzGerald. Groupoids on a skew lattice of objects. Art Discrete Appl. Math., 2(2):Paper No. 2.03,
+14, 2019.
+[51] D. G. FitzGerald and K. W. Lau. On the partition monoid and some related semigroups. Bull. Aust. Math.
+Soc., 83(2):273–288, 2011.
+[52] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.10.1, 2019.
+[53] V. Gould.
+Restriction and Ehresmann semigroups.
+In Proceedings of the International Conference on
+Algebra 2010, pages 265–288. World Sci. Publ., Hackensack, NJ, 2012.
+[54] V. Gould and C. Hollings. Restriction semigroups and inductive constellations. Comm. Algebra, 38(1):261–
+287, 2010.
+[55] V. Gould and Y. Wang. Beyond orthodox semigroups. J. Algebra, 368:209–230, 2012.
+[56] V. Gould and D. Yang.
+Every group is a maximal subgroup of a naturally occurring free idempotent
+generated semigroup. Semigroup Forum, 89(1):125–134, 2014.
+[57] R. Gray and N. Ruskuc. On maximal subgroups of free idempotent generated semigroups. Israel J. Math.,
+189:147–176, 2012.
+[58] R. Gray and N. Ruškuc. Maximal subgroups of free idempotent-generated semigroups over the full trans-
+formation monoid. Proc. Lond. Math. Soc. (3), 104(5):997–1018, 2012.
+[59] J. A. Green. On the structure of semigroups. Ann. of Math. (2), 54:163–172, 1951.
+[60] P. A. Grillet. The structure of regular semigroups. I. A representation. Semigroup Forum, 8(2):177–183,
+1974.
+[61] P. A. Grillet. The structure of regular semigroups. II. Cross-connections. Semigroup Forum, 8(3):254–259,
+1974.
+[62] T. Halverson and A. Ram. Partition algebras. European J. Combin., 26(6):869–921, 2005.
+[63] R. E. Hartwig. How to partially order regular elements. Math. Japon., 25(1):1–13, 1980.
+[64] J. B. Hickey. Semigroups under a sandwich operation. Proc. Edinburgh Math. Soc. (2), 26(3):371–382, 1983.
+[65] C. Hollings. Extending the Ehresmann-Schein-Nambooripad theorem. Semigroup Forum, 80(3):453–476,
+2010.
+[66] C. Hollings. The Ehresmann-Schein-Nambooripad theorem and its successors. Eur. J. Pure Appl. Math.,
+5(4):414–450, 2012.
+[67] C. D. Hollings and M. V. Lawson. Wagner’s theory of generalised heaps. Springer, Cham, 2017.
+[68] J. M. Howie. Fundamentals of semigroup theory, volume 12 of London Mathematical Society Monographs.
+New Series. The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications.
+[69] G. Huet. Confluent reductions: abstract properties and applications to term rewriting systems. J. Assoc.
+Comput. Mach., 27(4):797–821, 1980.
+[70] T. Imaoka. On fundamental regular ∗-semigroups. Mem. Fac. Sci. Shimane Univ., 14:19–23, 1980.
+[71] T. Imaoka. Prehomomorphisms on regular ∗-semigroups. Mem. Fac. Sci. Shimane Univ., 15:23–27, 1981.
+[72] T. Imaoka. Some remarks on fundamental regular ∗-semigroups. In Recent developments in the algebraic,
+analytical, and topological theory of semigroups (Oberwolfach, 1981), volume 998 of Lecture Notes in Math.,
+pages 270–280. Springer, Berlin, 1983.
+[73] P. R. Jones. A common framework for restriction semigroups and regular ∗-semigroups. J. Pure Appl.
+Algebra, 216(3):618–632, 2012.
+[74] V. F. R. Jones. Index for subfactors. Invent. Math., 72(1):1–25, 1983.
+[75] V. F. R. Jones. Hecke algebra representations of braid groups and link polynomials. Ann. of Math. (2),
+126(2):335–388, 1987.
+100
+
+[76] V. F. R. Jones. The Potts model and the symmetric group. In Subfactors (Kyuzeso, 1993), pages 259–267.
+World Sci. Publ., River Edge, NJ, 1994.
+[77] V. F. R. Jones. A quotient of the affine Hecke algebra in the Brauer algebra. Enseign. Math. (2), 40(3-
+4):313–344, 1994.
+[78] L. H. Kauffman. State models and the Jones polynomial. Topology, 26(3):395–407, 1987.
+[79] L. H. Kauffman. An invariant of regular isotopy. Trans. Amer. Math. Soc., 318(2):417–471, 1990.
+[80] M. Khoshkam and G. Skandalis. Regular representation of groupoid C∗-algebras and applications to inverse
+semigroups. J. Reine Angew. Math., 546:47–72, 2002.
+[81] N. V. Kitov and M. V. Volkov.
+Identities of the Kauffman monoid K4 and of the Jones monoid J4.
+In Fields of logic and computation. III, volume 12180 of Lecture Notes in Comput. Sci., pages 156–178.
+Springer, Cham, [2020.
+[82] G. Kudryavtseva and V. Mazorchuk.
+On presentations of Brauer-type monoids.
+Cent. Eur. J. Math.,
+4(3):413–434 (electronic), 2006.
+[83] M. Laca, I. Raeburn, J. Ramagge, and M. F. Whittaker. Equilibrium states on operator algebras associated
+to self-similar actions of groupoids on graphs. Adv. Math., 331:268–325, 2018.
+[84] D. Larsson.
+Combinatorics on Brauer-type semigroups.
+U.U.D.M. Project Report 2006:3, Available at
+http://www.diva-portal.org/smash/get/diva2:305049/FULLTEXT01.pdf.
+[85] K. W. Lau and D. G. FitzGerald. Ideal structure of the Kauffman and related monoids. Comm. Algebra,
+34(7):2617–2629, 2006.
+[86] M. V. Lawson. Semigroups and ordered categories. I. The reduced case. J. Algebra, 141(2):422–462, 1991.
+[87] M. V. Lawson. Inverse semigroups. World Scientific Publishing Co., Inc., River Edge, NJ, 1998. The theory
+of partial symmetries.
+[88] M. V. Lawson. Ordered groupoids and left cancellative categories. Semigroup Forum, 68(3):458–476, 2004.
+[89] M. V. Lawson. Non-commutative Stone duality: inverse semigroups, topological groupoids and C∗-algebras.
+Internat. J. Algebra Comput., 22(6):1250058, 47, 2012.
+[90] M. V. Lawson. Recent developments in inverse semigroup theory. Semigroup Forum, 100(1):103–118, 2020.
+[91] M. V. Lawson. On Ehresmann semigroups. Semigroup Forum, 103(3):953–965, 2021.
+[92] G. Lehrer and R. Zhang. The second fundamental theorem of invariant theory for the orthogonal group.
+Ann. of Math. (2), 176(3):2031–2054, 2012.
+[93] G. Lehrer and R. B. Zhang.
+The Brauer category and invariant theory.
+J. Eur. Math. Soc. (JEMS),
+17(9):2311–2351, 2015.
+[94] X. Li. Semigroup C∗-algebras and amenability of semigroups. J. Funct. Anal., 262(10):4302–4340, 2012.
+[95] X. Li. Partial transformation groupoids attached to graphs and semigroups. Int. Math. Res. Not. IMRN,
+(17):5233–5259, 2017.
+[96] S. Mac Lane.
+Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics.
+Springer-Verlag, New York, second edition, 1998.
+[97] V. Maltcev and V. Mazorchuk. Presentation of the singular part of the Brauer monoid. Math. Bohem.,
+132(3):297–323, 2007.
+[98] P. Martin. Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction.
+J. Knot Theory Ramifications, 3(1):51–82, 1994.
+[99] P. Martin. The structure of the partition algebras. J. Algebra, 183(2):319–358, 1996.
+[100] P. Martin and V. Mazorchuk.
+On the representation theory of partial Brauer algebras.
+Q. J. Math.,
+65(1):225–247, 2014.
+[101] P. P. Martin. The decomposition matrices of the Brauer algebra over the complex field. Trans. Amer. Math.
+Soc., 367(3):1797–1825, 2015.
+[102] P. P. Martin and G. Rollet. The Potts model representation and a Robinson-Schensted correspondence for
+the partition algebra. Compositio Math., 112(2):237–254, 1998.
+[103] V. Mazorchuk. Endomorphisms of Bn, PBn, and Cn. Comm. Algebra, 30(7):3489–3513, 2002.
+[104] J. Meakin, P. A. Azeef Muhammed, and A. R. Rajan. The mathematical work of K. S. S. Nambooripad.
+In Semigroups, categories, and partial algebras, volume 345 of Springer Proc. Math. Stat., pages 107–140.
+Springer, Singapore, 2021.
+[105] D. Milan and B. Steinberg. On inverse semigroup C∗-algebras and crossed products. Groups Geom. Dyn.,
+8(2):485–512, 2014.
+101
+
+[106] D. D. Miller and A. H. Clifford. Regular D-classes in semigroups. Trans. Amer. Math. Soc., 82:270–280,
+1956.
+[107] J. D. Mitchell et al. Semigroups - GAP package, Version 3.3.1, 2020.
+[108] H. Mitsch. A natural partial order for semigroups. Proc. Amer. Math. Soc., 97(3):384–388, 1986.
+[109] W. D. Munn. Fundamental inverse semigroups. Quart. J. Math. Oxford Ser. (2), 21:157–170, 1970.
+[110] K. S. S. Nambooripad. Structure of regular semigroups. I. Mem. Amer. Math. Soc., 22(224):vii+119, 1979.
+[111] K. S. S. Nambooripad. The natural partial order on a regular semigroup. Proc. Edinburgh Math. Soc. (2),
+23(3):249–260, 1980.
+[112] K. S. S. Nambooripad and F. Pastijn. Subgroups of free idempotent generated regular semigroups. Semi-
+group Forum, 21(1):1–7, 1980.
+[113] K. S. S. Nambooripad and F. J. C. M. Pastijn. Regular involution semigroups. In Semigroups (Szeged,
+1981), volume 39 of Colloq. Math. Soc. János Bolyai, pages 199–249. North-Holland, Amsterdam, 1985.
+[114] T. E. Nordahl and H. E. Scheiblich. Regular ∗-semigroups. Semigroup Forum, 16(3):369–377, 1978.
+[115] F. Pastijn. The biorder on the partial groupoid of idempotents of a semigroup. J. Algebra, 65(1):147–187,
+1980.
+[116] A. L. T. Paterson. Groupoids, inverse semigroups, and their operator algebras, volume 170 of Progress in
+Mathematics. Birkhäuser Boston, Inc., Boston, MA, 1999.
+[117] A. L. T. Paterson. Graph inverse semigroups, groupoids and their C∗-algebras. J. Operator Theory, 48(3,
+suppl.):645–662, 2002.
+[118] M. Petrich. Certain varieties of completely regular ∗-semigroups. Boll. Un. Mat. Ital. B (6), 4(2):343–370,
+1985.
+[119] L. Pontrjagin. The general topological theorem of duality for closed sets. Ann. of Math. (2), 35(4):904–914,
+1934.
+[120] L. Pontrjagin. The theory of topological commutative groups. Ann. of Math. (2), 35(2):361–388, 1934.
+[121] G. B. Preston. Inverse semi-groups. J. London Math. Soc., 29:396–403, 1954.
+[122] G. B. Preston. Representations of inverse semi-groups. J. London Math. Soc., 29:411–419, 1954.
+[123] H. A. Priestley. Representation of distributive lattices by means of ordered stone spaces. Bull. London
+Math. Soc., 2:186–190, 1970.
+[124] H. A. Priestley. Ordered topological spaces and the representation of distributive lattices. Proc. London
+Math. Soc. (3), 24:507–530, 1972.
+[125] S. Roberts. King of infinite space. Walker and Company, New York, 2006. Donald Coxeter, the man who
+saved geometry, With a foreword by Douglas R. Hofstadter.
+[126] H. E. Scheiblich. Projective and injective bands with involution. J. Algebra, 109(2):281–291, 1987.
+[127] B. M. Schein. On the theory of generalized groups. Dokl. Akad. Nauk SSSR, 153:296–299, 1963.
+[128] B. M. Schein. On the theory of generalized groups and generalized heaps. In Theory of Semigroups and
+Appl. I (Russian), pages 286–324. Izdat. Saratov. Univ., Saratov, 1965. English translation in Twelve papers
+in logic and algebra, Amer. Math. Soc. Translations Ser 2 113, AMS, 1979, pp. 89–122.
+[129] M. H. Stone. The theory of representations for Boolean algebras. Trans. Amer. Math. Soc., 40(1):37–111,
+1936.
+[130] H. N. V. Temperley and E. H. Lieb. Relations between the “percolation” and “colouring” problem and other
+graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation”
+problem. Proc. Roy. Soc. London Ser. A, 322(1549):251–280, 1971.
+[131] V. V. Wagner. Generalized groups. Doklady Akad. Nauk SSSR (N.S.), 84:1119–1122, 1952.
+[132] V. V. Wagner. The theory of generalized heaps and generalized groups. Mat. Sbornik N.S., 32(74):545–632,
+1953.
+[133] S. Wang. An Ehresmann-Schein-Nambooripad-type theorem for a class of P-restriction semigroups. Bull.
+Malays. Math. Sci. Soc., 42(2):535–568, 2019.
+[134] S. Wang. An Ehresmann-Schein-Nambooripad theorem for locally Ehresmann P-Ehresmann semigroups.
+Period. Math. Hungar., 80(1):108–137, 2020.
+[135] Y. Wang. Beyond regular semigroups. Semigroup Forum, 92(2):414–448, 2016.
+[136] H. Wenzl. On the structure of Brauer’s centralizer algebras. Ann. of Math. (2), 128(1):173–193, 1988.
+[137] S. Wilcox. Cellularity of diagram algebras as twisted semigroup algebras. J. Algebra, 309(1):10–31, 2007.
+[138] M. Yamada.
+On the structure of fundamental regular ∗-semigroups.
+Studia Sci. Math. Hungar., 16(3-
+4):281–288, 1981.
+[139] M. Yamada. p-systems in regular semigroups. Semigroup Forum, 24(2-3):173–187, 1982.
+[140] D. Yang, I. Dolinka, and V. Gould. Free idempotent generated semigroups and endomorphism monoids of
+free G-acts. J. Algebra, 429:133–176, 2015.
+102
+