sign in sign up
<<
Home , Band (algebra), Empty semigroup, Free monoid, Identity element, Involution (mathematics)

arXiv:1505.02384v2 [math.GR] 15 Sep 2015 te hwdtemnmlsz fsc eiru s7 n tha and 7, is a such of size minimal the showed (they osnt Winker not. does qainltere;oeo hs nouin ed oareg a to cont leads involutions the these in of ) one diagram theories; related equational (and monoids partition omtn nouin nasmsml leri ru (suc SL( group algebraic linear special semisimple the a on involutions Commuting cebih[3 aeeape fbns(dmoetsemigro ( bands regular of non-isomorphic examples gave [93] Scheiblich oewl-nw lershv utpeivltosdefinea maps involutions multiple have and well-known 25–27,3 Some [6,7, monoids diagram several example) (for models prtosbitit hi intr,icuigteclass the including [43], signature, algebras their into built operations GL( w nouin ie,temap the (i.e., involutions two α iha neligascaiebnr prto)i funct a is operation) binary associative underlying an with rus[52,45,36,810.A itra)involu (internal) An non- gene l algebras special as [15,20,44,52,63,64,78,100]. groups, groups well class mapping as inv Groups, () 97], Coxeter 90, have 86, well-known 82, that 79, 67, algebras 60, 50, are 47, 43, There mathematics . of in branches role many in ubiquitous are Involutions Introduction 1 ( xy ,F n, = ) safunction a is Sym( osdrteqeto fwihgop r smrhcto isomorphic are groups which of question the consider ,wihcnit falinvertible all of consists which ), α nivlto nasemigroup a on An S:2M5 02;2B7 20M20. 20B27; 20B25; 20M15; MSC: Keywords I ( ( S S y nteset the on ) falsc nouin on involutions such all of ) ) etefrRsac nMteais colo optn,En Computing, of School Mathematics; in Research for Centre α C ngop eeae yivltoso semigroup a of involutions by generated groups On ( x ∗ ,frevery for ), agba 3 6 7 7,M-rus[] n regular and [8], MI-groups 97], 37, 36, [3, -algebras nvriyo etr yny okdBg19,PnihNW2 NSW Penrith 1797, Bag Locked Sydney, Western of University eirus nouin,atmrhss anti-. automorphisms, involutions, , : α eateto scityadBhvoa cecs Imaging Sciences; Behavioral and Psychiatry of Department tal. et : a o dShilc nteocso fhs7t birthday. 77th his of occasion the on Scheiblich Ed For A S htsatisfies that S ∗ nvriyo aionaDvs ai,C 51,Uie Sta United 95616, CA Davis, Davis, California of University → 9]gv xmlso eiruswt niatmrhssb anti-automorphisms with semigroups of examples gave [99] eivsiaetegroups the investigate We . 7→ smgop.Auinger -semigroups. ,F n, S A ,y x, htsatisfies that − ) 1 A ∈ ⊆ and 7→ S GL( a S ayvreiso eirusadagba aeinvolutory have algebras and semigroups of varieties Many . 2 S ( o n ler iha neligascaiebnr operation) binary associative underlying an with algebra any (or A ) n(xenl nouino semigroup a on involution (external) An 1). = A ,F n, eeae a generates eodh ucdavis.edu @ tenordahl hmsE Nordahl E. Thomas T 7→ α ) [email protected] @ J.East − n ( )yeda yield )) xy 1 A ae East James × T = ) ( = Abstract tal. et n ohdfieivltoso h eea iergroup linear general the on involutions define both arcsoe field a over matrices A 1 C α − ( ( y Z S 1 6 tde w ieetivltoso h finite the on involutions different two studied [6] ) ) 2 o eti lse fsemigroups of classes certain for ) α T × ini ru sa lmn fodr2(.. a (i.e., 2 of element an is group a in tion ( so rus nes eirus[7,cellular [67], semigroups inverse groups, of es sanninratmrhs fGL( of non-inner a is ) x C Z and ) ae y(nenl nouin uha the as such involutions (internal) by rated 2 ( lto prtr endo hm[,8 17, 8, [3, them on defined operators olution ion na rus n o-bla nt simple finite non-abelian and groups, inear ,9 57,76]. 65,72, 0,39, n aepae atclrysignificant particularly a played have and , p)frwihtoivltosgv ieto rise give involutions two which for ups) S ular gaigo h soitdLeagba[84]. algebra Lie associated the on -grading C o utbesemigroup suitable a for ) l nte.Freape h inverse the example, For them. on ble steivreadtasoemp on maps transpose and inverse the as h ( S α x f(neety o-ntl based non-finitely (inherently) of ext α = ) htteeaefu uhsemigroups such four are there that t ∗ ( : iern n Mathematics and gineering α smgopsrcueadteother the and structure -semigroup S ( ∗ h x I smgop 8] h atrclass latter the [82]; -semigroups → )= )) eerhCenter Research ( S F ) S 5,Australia 751, i h opsto fthese of composition The . x ftesmercgroup symmetric the of satisfying tes o all for ,y x, S α S tn involution no ut S ( . ∈ o n algebra any (or α n also and , ( S x The . )= )) x unary ,F n, and ). of minimal size, all of which are 3-nilpotent). Ciobanu et al. [13] made crucial use of a free with involution in their work on word equations in free groups. Bacovsk´y[8] investigated a class of monoids with involution that have applications in processor networks and fuzzy numbers. Gustafson et al. [44] showed that any square of determinant ±1 over any field is the product of (at most) four involutions; a square matrix is the product of two involutions if and only if it is invertible and similar to its own inverse [20,100]. A linear bound for products of involutory matrices was given in [57]. Everitt and Fountain studied partial mirror symmetries by investigating certain factorizable inverse monoids generated by “partial reflections” of a space [34, 35]; see also [29]. Lusztig and Vogan investigated certain actions on Hecke algebras via involutions on Coxeter groups that permute the simple reflections [70, 71]. Francis et al. made extensive use of involution-generated groups in their work on algebraic bacterial genomics [1, 9, 31, 32]. Finally, we must mention the vast body of work of authors such as Dolinka, Imaoka, Jones, Petrich, Reilly, Scheiblich and Yamada [16, 21–24, 53–56, 60, 61,85,88,91–93,101] on varieties of involution semigroups and bands; see especially [85] for a discussion of early work on this topic.

In this work, we are interested in all the involutions on a semigroup S, and we study these by defining a group C (S) as follows. First, we let I(S) be the set of all involutions on S. So I(S) is a (possibly empty) of Sym(S), the on S (which consists of all of S). We then define C (S)= hI(S)i to be the subgroup of Sym(S) generated by I(S). (We interpret C (S)= h∅i = {idS} to be the trivial subgroup of Sym(S) in the case that S has no involutions definable on it; this occurs for left- or right-zero semigroups, for example. Throughout, we write idX for the identity mapping on the set X, or just id if the set is clear from context.)

As an example, consider the Klein 4-group, K = Z2 × Z2. If we denote by x,y,z the non-identity elements of K, then one easily checks that any of x,y,z induces an automorphism of K. In particular, since K is commutative, the transpositions (x,y), (x, z) and (y, z) each induce involutions of K. It quickly follows that C (K) = Aut(K) is isomorphic to Sym(3), the symmetric group on three letters. For examples where C (S) does not coincide with Aut(S), see Sections 4 and 5. In fact, it is possible for non-isomorphic semigroups S =6∼ T to have Aut(S) =∼ Aut(T ) but C (S) =6∼ C (T ), so the C (S) invariant distinguishes between some semigroups that the Aut(S) invariant does not (see Sections 4 and 5.5). The main purpose of this paper is to investigate the two most natural questions to ask:

Question 1. Given a semigroup S, can we describe the group C (S)?

Question 2. Given a group G generated by (internal) involutions, does there exist a semigroup S such that G =∼ C (S)?

We answer Question 1 for several families of semigroups including: free semigroups and free commuta- tive semigroups of arbitrary rank; free groups and free abelian groups of finite rank; (finite or infinite) full transformation semigroups, symmetric and dual symmetric inverse monoids; finite partition mon- oids; semigroups of complex matrices; rectangular bands; and graph semigroups.

We also utilise a construction due to Ara´ujo et al. [2] to give an affirmative answer to Question 2. Specifically, we show that any involution generated group is isomorphic to C (S) for some commutative semigroup S. However, none of the involutions on such a commutative semigroup S are “proper”, in the sense that they only “reverse the operation” of S (i.e., satisfy α(xy)= α(y)α(x)) because they preserve it. Since many semigroups of interest (such as non-commutative groups, inverse semigroups and regular ∗-semigroups) do have proper involutions, we therefore consider the following more focussed question to be of paramount importance.

Question 3. Given a group G generated by (internal) involutions, does there exist a semigroup S with a proper involution such that G =∼ C (S)?

(As alluded to above, we say an involution is proper if it is not a .) Our investigation of graph semigroups will lead us to a partial answer to this question. Namely, we show that for

2 any involution-generated group G, there exists a semigroup S with a proper involution such that C (S) =∼ G × Z2. We also show that if G =∼ C (S) for some semigroup S with a proper involution, then G =∼ N ⋊Z2 for some normal subgroup N of G; in particular, this shows that Question 3 has a negative answer for some involution-generated groups.

Acknowledgements. We thank Dr Heiko Dietrich for several helpful conversations during the prepa- ration of this article, and also Prof Mohan Putcha for encouragement and advice.

2 Preliminaries

In this section, we fix notation, and gather together various facts that we will need in our investigations.

Let S and T be semigroups. Recall that a α : S → T is a homomorphism (resp., anti- homomorphism) if α(xy)= α(x)α(y) (resp., α(xy)= α(y)α(x)) for all x,y ∈ S. An anti-homomorphism is called proper if it is not also a homomorphism (in which case T must not be commutative). An (anti-)homomorphism is called an (anti-)isomorphism if it is bijective. The inverse mapping α−1 : T → S of an (anti-)isomorphism α : S → T is also an (anti-)isomorphism. An (anti-)isomorphism S → S is called an (anti-)automorphism.

We write Aut(S) (resp., Aut−(S)) for the set of all automorphisms (resp., anti-automorphisms) of the semigroup S. We write Aut±(S) = Aut(S) ∪ Aut−(S). Note that while Aut(S) always contains the identity mapping, Aut−(S) may be empty (this is true of left- or right-zero semigroups, for example, and for the full TX when |X| ≥ 2; see Section 5.5). The composite of two automorphisms or two anti-automorphisms is an automorphism, while the composite of an automor- phism with an anti-automorphism (in any order) is an anti-automorphism. So Aut±(S) is a group under composition, which we call the signed automorphism group of S.

An involution of S is an anti-automorphism α of order 2; that is, α ∈ Aut−(S) and α2 = id 6= α. We write I(S) for the set of all involutions of S, and C (S)= hI(S)i for the subgroup of Aut±(S) generated by I(S). Note that C (S)= {idS} if I(S)= ∅. The proof of the next result is obvious, and is omitted.

Lemma 2.1. Let S be a semigroup.

(i) If S is commutative, then Aut−(S) = Aut(S) = Aut±(S).

(ii) If S is not commutative, then either Aut−(S)= ∅ or else the map

− Φβ : Aut(S) → Aut (S) : α 7→ αβ

is a for any β ∈ Aut−(S), and we have a short exact :

± {1}→ Aut(S) → Aut (S) → Z2 → {1}. That is, Aut(S) is a (normal) subgroup of index at most 2 of Aut±(S). ✷

So if S is not commutative and Aut−(S) is non-empty, then the signed automorphism group Aut±(S) is an extension of Z2 by Aut(S). This extension splits if and only if I(S) 6= ∅. Non-commutative semigroups with I(S) = ∅= 6 Aut−(S) are described in [99], where it is shown that the smallest such semigroup has size 7.

Recall that an atom of a semigroup S (which may or may not be a monoid) is an element a ∈ S that cannot be factorized as a product of non-identity elements of S. Write A(S) for the set of all atoms of S (which may be empty).

Lemma 2.2. Let α : S → T be an (anti-)isomorphism. Then α(A(S)) = A(T ).

3 Proof. We just prove the statement for anti-isomorphisms. Let a ∈ A(S). If α(a)= bc for some non- identity elements b, c ∈ T , then a = α−1(bc) = α−1(c)α−1(b), where α−1(c), α−1(b) are non-identity elements of S, contradicting the assumption that a ∈ A(S). (Note that S is a monoid if and only if T is a monoid, in which case α must map the identity of S to the identity of T .) It follows that α(a) ∈ A(T ), so α(A(S)) ⊆ A(T ). Applying this also to the anti-isomorphism α−1 : T → S, we obtain A(T )= α(α−1(A(T ))) ⊆ α(A(S)) ⊆ A(T ), giving α(A(S)) = A(T ). ✷

We write Sym(X) for the symmetric group on a set X, which consists of all permutations of X. In what follows, we will frequently use the fact that a symmetric group Sym(X) is generated by (internal) involutions. Although this is certainly well known, especially in the finite case (with finite Sym(X) being a of Type A), we include a short proof for convenience. (It is also known that any element of a finite Coxeter group is either an involution or the product of two involutions [11].)

Lemma 2.3. Let π ∈ Sym(X), where X is an arbitrary set. Then π = στ for some σ, τ ∈ Sym(X) with σ2 = τ 2 = id.

Proof. If π = id, then the result is clear, so suppose otherwise. Write π = i∈I γi, where I is some indexing set, and each γi is a non-trivial cycle of some subset Xi ⊆ X, with the Xi pairwise disjoint Q (and each of size at least 2). Note that it is possible for I to be infinite (in which case π is simply the “formal product” of the γi), and for some of the Xi to be (countably) infinite. Fix some i ∈ I. We consider three cases.

Case 1. If γi = (x1,...,x2k+1) for some k ≥ 1, then γi = σiτi, where

σi = (x2,x2k+1)(x3,x2k) · · · (xk+1,xk+2) and τi = (x1,x2k+1)(x2,x2k) · · · (xk,xk+2).

Case 2. If γi = (x1,...,x2k) for some k ≥ 1, then γi = σiτi, where

σi = (x2,x2k)(x3,x2k−1) · · · (xk,xk+2) and τi = (x1,x2k)(x2,x2k−1) · · · (xk,xk+1).

Case 3. If γi = (...,x−2,x−1,x0,x1,x2,...) is an infinite cycle, then γi = σiτi, where

σi = (x1,x−1)(x2,x−2)(x3,x−3) · · · and τi = (x0,x−1)(x1,x−2)(x2,x−3) · · · . 2 2 ✷ Finally, we see that π = στ, where σ = i∈I σi and τ = i∈I τi satisfy σ = τ = id. Q Q We conclude this section with a simple group theoretic result. As usual, if G is a group, we denote by [G, G] the (normal) subgroup of G generated by all commutators [g, h]= ghg−1h−1 with g, h ∈ G.

−1 Lemma 2.4. Let G be a group, and denote by KG the subgroup of G×G generated by {(g, g ) : g ∈ G}. Then KG = (g, h) ∈ G × G : gh ∈ [G, G] .

In particular, KG is an extension of Gby [G, G].

Proof. Put Ω = (g, h) ∈ G × G : gh ∈ [G, G] . First note that for any g, h ∈ G, KG contains (g, g−1)(hg−1, gh−1)(h−1, h) = (ghg−1h−1, 1), where 1 denotes the of G. Similarly, −1 −1  (1, ghg h ) ∈ KG. It follows that [G, G] × [G, G] ⊆ KG. So if g, h ∈ G satisfy gh ∈ [G, G], then −1 (g, h) = (gh, 1)(h , h) ∈ KG, showing that Ω ⊆ KG. Conversely, we show by induction that

−1 −1 −1 −1 (g1 · · · gk, g1 · · · gk ) = (g1, g1 ) · · · (gk, gk ) ∈ Ω for all g1,...,gk ∈ G. (2.5) If k ≤ 2, this is clear, so suppose k ≥ 3. Then

−1 −1 −1 −1 −1 −1 −1 g1 · · · gk · g1 · · · gk = g1 · · · gk−2(gk−1gk) · g1 · · · gk−2(gk−1gk) · gk−1gkgk−1gk .   4 −1 −1 By an induction hypothesis, the bracketed term belongs to [G, G], giving g1 · · · gk · g1 · · · gk ∈ [G, G]. This completes the proof of (2.5), and shows that KG ⊆ Ω. Finally, we note that the epimorphism KG → G : (g, h) 7→ g has {1}× [G, G]. ✷

Remark 2.6. The group KG may be thought of as the kernel of the epimorphism G × G → G/[G, G] sending the pair (g, h) to the coset [G, G]gh. This construction may of course be generalised by replacing G × G with the of n ≥ 2 copies of G. The resulting groups, denoted K(G, n), are studied in [68] in connection to invariants for complex algebraic surfaces (among other things).

Example 2.7. It is well known that

Alt(X) if |X| < ℵ [Sym(X), Sym(X)] = 0 (Sym(X) if |X|≥ℵ0. (In fact, Ore [83] showed that any element of [Sym(X), Sym(X)] is itself a commutator, and not just a product of commutators.) It follows that

{(σ, τ) ∈ Sym(X) × Sym(X) : στ ∈ Alt(X)} if |X| < ℵ0 KSym(X) = (Sym(X) × Sym(X) if |X|≥ℵ0. Consider now the case in which X is finite and put

K = KSym(X) = {(σ, τ) ∈ Sym(X) × Sym(X) : στ ∈ Alt(X)}. So Alt(X) × Alt(X) ≤ K ≤ Sym(X) × Sym(X). (Here, ≤ denotes the subgroup .) In fact, we have an internal decomposition K = (Alt(X) × Alt(X)) ⋊ h(π,π)i, where π is any fixed odd permutation of order 2 (such as a transposition), but we note that this is not a wreath product. We also note that the condition στ ∈ Alt(X) is equivalent to saying that σ and τ have the same parity.

3 Obtaining any involution-generated group as C (S)

Ara´ujo et al. [2] gave a number of constructions that realise any finite group as the automorphism group of various kinds of finite semigroups. Here we review one of these constructions and show how it applies to the present considerations.

Graphs play an important role in this section and the next, so we take this opportunity to fix our notation. Throughout, graphs are assumed to be undirected and have no loops or edges. Let Γ be a graph with vertex and edge sets X and E, respectively. Recall that an automorphism of Γ is a bijection π : X → X such that {x,y} ∈ E if and only if {π(x),π(y)} ∈ E. We write Aut(Γ) for the group of automorphisms of Γ. Note that Aut(Γ) is a subgroup of Sym(X), the symmetric group over X.

Let G be an arbitrary group. It is well known that G is isomorphic to the automorphism group Aut(Γ) of a suitable graph Γ. This was originally proved for finite groups (in which case the graph may be taken to be finite) by Frucht [41], and then extended to infinite groups by de Groot [18]. Various strengthenings of this result exist, including to graphs of given chromatic number or satisfying various connectivity or regularity conditions; see for example [42, 59, 89]. But all we need to know is that Γ may be chosen so that it has at least one edge (and it is easy to see why this is possible, assuming de Groot’s theorem). We fix such a graph Γ, with G =∼ Aut(Γ), and write X and E for the vertex and edge set of Γ, respectively. As in [2], we define a semigroup S = X ∪ {Y,N}, where Y and N are new symbols that do not belong to X, and with defined, for u, v ∈ S, by

Y if u, v ∈ X and {u, v}∈ E uv = (N otherwise.

5 Note that S is commutative and 3-nilpotent (i.e., N is a and abc = N for any a, b, c ∈ S). It is easy to check that: (i) any (graph) automorphism α ∈ Aut(Γ) extends to a (semigroup) automorphismα ˆ ∈ Aut(S) by further definingα ˆ(Y )= Y andα ˆ(N)= N; and

(ii) any β ∈ Aut(S) must be of the formα ˆ for some α ∈ Aut(Γ). (The assumption that Γ has an edge is necessary to prove that β(Y )= Y .) Thus, Aut(S) =∼ Aut(Γ) and it follows that G =∼ Aut(S). In fact, since S is commutative, it follows that Aut±(S) = Aut(S) =∼ G. In particular, if G is an involution-generated group, then so too is Aut±(S) =∼ G, so that C (S) = Aut±(S) =∼ G. This therefore proves the following.

Theorem 3.1. If G is an arbitrary involution-generated group, then G =∼ C (S) for some (commutative) semigroup S. ✷

Theorem 3.1 gives an affirmative answer to Question 2 from the introduction. However, since the semi- group S constructed above is commutative, S has no proper involutions. So the above considerations do not say anything about Question 3, which we restate here for convenience.

Question 3. Given a group G generated by (internal) involutions, does there exist a semigroup S with a proper involution such that G =∼ C (S)?

Since many well-studied involutory semigroups (such as non-commutative groups, inverse semigroups, regular ∗-semigroups, and so on) have proper involutions, Question 3 does seem a natural one to investigate. The next result shows that there are some restrictions on the kind of group G that may be isomorphic to C (S) for a semigroup S with a proper involution; namely, G must be a split extension of Z2 by some normal subgroup.

Proposition 3.2. Let S be a semigroup with a proper involution ι. Then we have (internal) semidirect product decompositions

Aut±(S) = Aut(S) ⋊ hιi and C (S) = (C (S) ∩ Aut(S)) ⋊ hιi.

± That is, Aut (S) (resp., C (S)) is a split extension of Z2 by Aut(S) (resp., C (S) ∩ Aut(S)).

Proof. This follows quickly from the facts that Aut(S) (resp., C (S) ∩ Aut(S)) is an index 2 subgroup of Aut±(S) (resp., C (S)), and that ι has order 2. ✷

Remark 3.3. It follows immediately that the class of groups isomorphic to C (S) for some semigroup S with a proper involution is smaller than the class of all involution-generated groups. For example, if G is a finite non-abelian simple group, then G is generated by its elements of order 2 (since |G| is even, and the subgroup generated by elements of order 2 is a non-trivial normal subgroup of G). But such a group G is obviously not a split extension.

We now investigate a special case in which the semidirect product decompositions in Proposition 3.2 are direct products. For a semigroup S, write J(S)= {α ∈ Aut(S) : α2 = id}, and put G (S)= hJ(S)i. (Note that id ∈ J(S) for all S.)

Proposition 3.4. Let S be a semigroup with a proper involution ι that commutes with all automor- phisms of S. Then Ψι : J(S) → I(S) : α 7→ αι is a bijection, and we have (internal) direct product decompositions

Aut±(S) = Aut(S) × hιi and C (S)= G (S) × hιi.

6 Proof. If α ∈ J(S), then αι ∈ Aut−(S) and (αι)2 = α2ι2 = id, so that αι ∈ I(S). Conversely, if β ∈ I(S), then β = (βι)ι, with βι ∈ Aut(S) and (βι)2 = id. To complete the proof, it suffices to show that C (S) ∩ Aut(S) = G (S). If α ∈ C (S) ∩ Aut(S), then α = (β1ι) · · · (β2kι) for some β1,...,β2k ∈ J(S), from which it follows that α = β1 · · · β2k ∈ G (S). Conversely, if α ∈ G (S), then α = β1 · · · βl for some β1,...,βl ∈ J(S) where we may assume that l is even (since id ∈ J(S)), in which case, α = (β1ι) · · · (βlι) ∈ C (S). ✷

In particular, Proposition 3.4 holds for non-commutative inverse semigroups (including non-abelian groups), since we may use the inverse mapping ι : S → S : s 7→ s−1, noting that α(s−1) = α(s)−1 for any semigroup homomorphism α : S → T between inverse semigroups. In the next section, we will study another class of semigroups (that are not inverse semigroups) that Proposition 3.4 applies to. As a consequence of our investigations, we will see that for any involution-generated group G, there exists a semigroup S with a proper involution such that C (S) =∼ Z2 × G.

Remark 3.5. In the argument used to prove Theorem 3.1, above, we made use of the fact [2] that every group is isomorphic to the automorphism group Aut(S) of some commutative semigroup S. It is known [19, 48] that some groups (such as the cyclic group of order 5) are not isomorphic to the automorphism group Aut(G) of any group G. So the question of which (involution-generated) groups are isomorphic to C (G) for some group G seems to be of importance. For example, we noted in the introduction that the symmetric group of degree 3 is isomorphic to C (K) where K = Z2 × Z2 is the Klein 4-group.

4 Graph semigroups

In this section, we calculate C (S) in the case that S belongs to the class of graph semigroups (see below for the definition). Part of the motivation for doing this is that the free semigroups and free commutative semigroups belong to this class. The other motivation is that we may use the results to give a partial answer to Question 3 above. (We note that the results of this section apply equally well to the corresponding monoids, but we just treat semigroups for convenience.)

We begin by fixing notation for dealing with semigroup presentations. Let X be an arbitrary set, and write X+ for the free semigroup over X, which consists of all non-empty words over X under the semigroup operation of . We regard X as a subset of X+ in the usual way (as the set of words of length 1). For R ⊆ X+ × X+, we write R♯ for the congruence on X+ generated by R. We then define hX | Ri = X+/R♯. So hX | Ri is the set of all R♯-classes of words over X under the induced ♯ + operation on equivalence classes. We denote the R -class of w ∈ X by [w]R. If S is a semigroup and φ : X → S is a function, then φ extends uniquely to a homomorphism + + + αφ : X → S and an anti-homomorphism βφ : X → S defined, for x1 · · · xk ∈ X , by

αφ(x1 · · · xk)= φ(x1) · · · φ(xk) and βφ(x1 · · · xk)= φ(xk) · · · φ(x1).

If αφ preserves R, in the sense that αφ(u)= αφ(v) for all (u, v) ∈ R, then φ extends to a homomorphism

+ γφ : hX | Ri→ S defined by γφ([w]R)= αφ(w) for w ∈ X .

Similarly, if βφ preserves R, then φ extends to an anti-homomorphism

+ δφ : hX | Ri→ S defined by δφ([w]R)= βφ(w) for w ∈ X .

Note that αφ = βφ and (if applicable) γφ = δφ if S is commutative. Now let Γ be a graph with vertex and edge sets X and E, respectively. (Note that X may be an infinite, even uncountable, set.) The graph semigroup on Γ (see [38]) is defined by the presentation

SΓ = hX | RΓi where RΓ = {(xy,yx) : {x,y}∈ E}.

7 + As special cases, note that if Γ has no edges, then SΓ = X , while if Γ = KX is the complete graph on X (which has every possible edge {x,y} with x,y ∈ X and x 6= y), then SΓ is the free commutative + ♯ semigroup over X, which we denote by X↔. For simplicity, we will write w = [w]RΓ for the RΓ-class of a word w ∈ X+. Note that graph semigroups are also known as free partially commutative semigroups or right angled Artin semigroups in the literature. The next result is obvious, and its proof is omitted.

Lemma 4.1. If x,y ∈ X, then xy = yx if and only if {x,y}∈ E. ✷

+ In particular, SΓ is commutative if and only if Γ = KX , in which case SΓ = X↔, as noted above.

Proposition 4.2. Let Γ be a graph on vertex set X, and let π ∈ Aut(Γ) be an arbitrary (graph) automorphism of Γ. Then π extends uniquely to a (semigroup) automorphism γπ ∈ Aut(SΓ) and an − anti-automorphism δπ ∈ Aut (SΓ) defined by

γπ(x1 · · · xk)= π(x1) · · · π(xk) and δπ(x1 · · · xk)= π(xk) · · · π(x1)

+ + for all words x1 · · · xk ∈ X . Further, any automorphism or anti-automorphism of X is of one of these forms (as appropriate); that is,

− Aut(SΓ)= {γπ : π ∈ Aut(Γ)} and Aut (SΓ)= {δπ : π ∈ Aut(Γ)}.

Proof. Again, we just prove the statement for auti-automorphisms. Let π ∈ Aut(Γ). Since {x,y}∈ E if and only if {π(x),π(y)} ∈ E, it follows that δπ is a well-defined anti-homomorphism, and it is easy to check that δπ−1 is its inverse mapping.

Conversely, let ε : SΓ → SΓ be an arbitrary anti-automorphism. By Lemma 2.2, ε maps A(SΓ) bijec- tively onto itself, and it is clear that A(SΓ) = {x : x ∈ X}. Define π ∈ Sym(X) by ε(x) = π(x). We claim that, in fact, π ∈ Aut(Γ). Indeed, for any x,y ∈ X, we have

{x,y}∈ E ⇔ xy = yx ⇔ ε(xy)= ε(yx) ⇔ ε(y)ε(x)= ε(x)ε(y) ⇔ π(y)π(x)= π(x)π(y) ⇔ {π(x),π(y)}∈ E, where the first and last step uses Lemma 4.1. This completes the proof that π ∈ Aut(Γ), and it is clear then that ε = δπ. ✷

Corollary 4.3. Let Γ be a graph on vertex set X. Then

− Aut(SΓ)= {γπ : π ∈ Aut(Γ)} =∼ Aut(Γ) and Aut (SΓ)= {δπ : π ∈ Aut(Γ)}.

± (i) If Γ= KX , then Aut (SΓ) = Aut(SΓ) =∼ Aut(Γ) = Sym(X). ± (ii) If Γ 6= KX , then Aut (SΓ) =∼ Z2 × Aut(Γ).

Proof. By Proposition 4.2, we have Aut(SΓ) = {γπ : π ∈ Aut(Γ)}. It is also clear that for any π,σ ∈ Aut(Γ), γπ = γσ if and only if π = σ, and that γπγσ = γπσ, giving Aut(SΓ) =∼ Aut(Γ). If + ± Γ = KX , then SΓ = X↔ is commutative, whence Aut (SΓ) = Aut(SΓ), so (i) then follows from the obvious fact that Aut(KX ) = Sym(X). For (ii), note that that for all π,σ ∈ Aut(Γ),

γπ = γσ ⇔ δπ = δσ ⇔ π = σ, γπγσ = δπδσ = γπσ, γπδσ = δπγσ = δπσ. ± It quickly follows that the map Z2 × Aut(Γ) → Aut (SΓ) given by (0,π) 7→ γπ and (1,π) 7→ δπ is an isomorphism. ✷

We now wish to use Corollary 4.3 to obtain information about the groups C (SΓ) = hI(SΓ)i. Since − 2 I(SΓ) ⊆ Aut (SΓ), and since δπ = γπ2 for π ∈ Aut(Γ), we see that 2 I(SΓ)= {δπ : π ∈ Aut(Γ), π = id}.

8 + + Note that I(SΓ) contains δid, which is induced by the word-reversing map X → X . (So part (ii) of the previous result also follows from Proposition 3.4, since δid clearly commutes with any automorphism γπ.) For a graph Γ, let us now write I(Γ) = {α ∈ Aut(Γ) : α2 = id 6= α} for the set of all involutions of Γ, and C (Γ) = hI(Γ)i for the subgroup of Aut(Γ) generated by these involutions. As an immediate consequence of Corollary 4.3 and Lemma 2.3 (which states that Sym(X) is generated by involutions), we have the following.

Theorem 4.4. Let Γ be a graph on vertex set X.

(i) If Γ= KX , then C (SΓ) =∼ Sym(X).

(ii) If Γ 6= KX , then C (SΓ) =∼ Z2 × C (Γ). ✷

Remark 4.5. In the special case that Γ has no edges, we have Aut(Γ) = Sym(X) and, since Sym(X) + is involution generated (Lemma 2.3), C (Γ) = Aut(Γ). We have already noted that SΓ = X in this + case, so we obtain C (X ) =∼ Z2 × Sym(X).

Remark 4.6. A much simpler construction yields a semigroup S with C (S) =∼ Sym(X). We define S = X ∪ {0}, where 0 is a symbol that does not belong to X, and simply declare uv = 0 for all u, v ∈ S. Then clearly C (S) = Aut±(S) = Aut(S) =∼ Sym(X).

As noted in Section 3, any group is isomorphic to the automorphism group of a suitable graph.

Theorem 4.7. Let G be a group generated by involutions. Then there exists a semigroup S with a proper involution such that C (S) =∼ Z2 × G. If G is finite, then S may be taken to be finitely generated.

Proof. Choose some graph Γ such that G =∼ Aut(Γ). We may assume that Γ is not complete and is finite if G is finite. Since G is generated by involutions, so too is Aut(Γ) =∼ G, so C (Γ) = Aut(Γ). The result then follows from Theorem 4.4(ii). ✷

5 Other families of semigroups

In this section, we calculate C (S) for various natural families of semigroups: namely, free groups and free abelian groups of finite rank; finite cyclic groups; symmetric groups; full transformation semigroups; symmetric and dual symmetric inverse monoids; finite diagram monoids; monoids of complex matrices; and rectangular bands. We also give a natural construction that allows one to embed a semigroup − − S with Aut (S) = ∅ into a semigroup DS with Aut (DS ) 6= ∅; indeed, the involutions of DS are in one-one correspondence with the automorphisms of S.

5.1 Free groups of finite rank

Let X = {x1,...,xn} be a finite set with |X| = n ≥ 2 (the n = 1 case is considered in the next section), and let FX be the over X. So elements of FX are equivalence classes of words over −1 −1 {x1,x1 ,...,xn,xn }, where two words are considered equivalent if one may be transformed into the other by successively using the free group relations

−1 −1 xixi = xi xi = 1 for each i,

9 where 1 denotes the empty word. Nielsen [81] gave a generating set for the automorphism group Aut(FX ). Specifically, it was shown that Aut(FX ) is generated by the automorphisms σ1,...,σn−1, α, β defined by

xi+1 if j = i −1 x1 if j = 1 x1x2 if j = 1 σi(xj)= xi if j = i + 1 α(xj )= β(xj)= (xj if j 6= 1, (xj if j 6= 1. xj if j 6= i, i + 1,

(Here we simplify notation by writing a word instead of its equivalence class.) Actually, in [81], the automorphisms σ2,...,σn−1 were replaced by the automorphism γ defined by γ(xj)= xj+1 for each j, where the subscript is read modulo n. But since {σ1,...,σn−1} and {σ1, γ} both generate the auto- of FX induced by permuting the elements of X, the generating set we use here is equivalent. k k Note that σ1,...,σn−1, α are all of order 2, but that β is of infinite order; indeed, β (x1) = x1x2 for k ≥ 1. However, it is easy to check that β = β1β2, where β1, β2 ∈ Aut(FX ) are defined by

x1x2 if j = 1 −1 −1 x2 if j = 2 β1(xj)= x2 if j = 2 β2(xj)= (xj if j 6= 2. xj if j ≥ 3,  Moreover, the automorphisms β1, β2 are of order 2, so it follows that Aut(FX ) is involution-generated. Combining this with Proposition 3.4, it follows that

± G (FX )= hJ(FX )i = Aut(FX ) and C (FX ) = Aut (FX )= hιi× Aut(FX ),

±1 ±1 where ι : FX → FX is the inversion map, which sends (the equivalence class of) the word xi1 · · · xik ∓1 ∓1 to xik · · · xi1 .

5.2 Free abelian groups of finite rank

A free of finite rank n ≥ 1 is the quotient of a free group of rank n by the relations declaring all generators to commute with each other; such a group is isomorphic to Zn. Since Zn is abelian, Aut±(Zn) = Aut(Zn) and C (Zn) = G (Zn). The automorphisms of Zn are precisely the invertible linear maps Zn → Zn, where Zn is considered as a (free) Z-. These are in one-one correspondence with the invertible n × n matrices over Z: that is, the matrices of determinant ±1. We denote the set of all such matrices by GL(n, Z). It is well known [45] that GL(n, Z) is generated by 2 involutions (i.e., matrices that satisfy A = In, the n × n ); Ishibashi [57] gave an upper bound on the number of involutions required to generate a given element of GL(n, Z) for n ≥ 3. It follows that C (Zn) = Aut(Zn) = GL(n, Z). Zn a b It is easy to check that there are infinitely many involutions on for n ≥ 2; for example, c −a is an involution of Z2 for any a, b, c ∈ Z satisfying a2 + bc = 1.   We note that free groups and free abelian groups have a common generalisation to graph groups GΓ, defined in an analogous way to the graph semigroups of Section 4. Generators for the automorphism groups Aut(GΓ) are given in [66,96], but a description of the involution-generated subgroup of Aut(GΓ) is beyond the scope of the present work.

5.3 Finite cyclic groups

Let n ≥ 2 be an integer, and write Zn = {0, 1,...,n−1} for the additive group of modulo n. A homomorphism α : Zn → Zn is uniquely determined by α(1) = k, where k ∈ Zn, and any such k gives rise to a homomorphism; so we denote this homomorphism by αk, and note that αk(m) = km for all

10 m ∈ Zn. Since αkαl = αkl, we see that End(Zm), the semigroup of all of Zn (i.e., all Zn → Zn), is isomorphic to (Zn, ·), the multiplicative semigroup of integers modulo n. The automorphism group Aut(Zn) is therefore isomorphic to Un = {m ∈ Zn : gcd(m,n) = 1}, the group 2 of multiplicative units of (Zn, ·). The involutions of Zn are therefore the maps αk with k = 1 6= k in Zn. Note that the composite of two such involutions is itself an involution or else the identity map. In other words, 2 C (Zn)= I(Zn) ∪ {id} = {αk : k ∈ Zn, k = 1}, where as usual id ∈ Aut(Zn) denotes the identity automorphism. Consider the prime factorization m m1 mr n = 2 p1 · · · pr (so m ≥ 0, p1,...,pr are distinct odd primes, and mi ≥ 1 for each i). It is well 2 R(n) known that the number of solutions to k =1 in Zn is equal to 2 , where

r if m ≤ 1 R(n)= r +1 if m = 2 r +2 if m ≥ 3.

(See for example [80], or [46] where involutions on finite abelian groups were studied.) Since every Z Z 2 Z ∼ ZR(n) element α ∈ C ( n)= I( n) ∪ {id} satisfies α = id, it follows that C ( n) = 2 .

5.4 Symmetric groups

Let X be an arbitrary set. It is well known [75, 95] that

Sym(X) if |X|= 6 2, 6 ∼ Aut(Sym(X)) = {id} if |X| = 2  Sym(X) ⋊Z2 if |X| = 6.

If |X| 6= 2, 6, then every automorphism of Sym( X) is inner; that is, if α ∈ Aut(Sym(X)), then there exists some π ∈ Sym(X) such that α(σ) = πσπ−1 for all σ ∈ Sym(X). When |X| = 6, there are non- inner automorphisms sending transpositions to products of three disjoint transpositions, resulting in Aut(Sym(X)) =∼ Sym(X)⋊Z2; for a nice recent exposition of this fact, see [77]. Since Sym(X) is a group ± and is generated by its involutions, Proposition 3.4 gives C (Sym(X)) = Aut (Sym(X)) =∼ Z2 ×Sym(X) ± for |X|= 6 2, 6. If |X| = 6, then C (Sym(X)) = Aut (Sym(X)) =∼ Z2 × (Sym(X) ⋊Z2). (If |X| = 2, then C (Sym(X)) = Aut±(Sym(X)) = {id}.)

5.5 Full transformation semigroups

The full transformation semigroup on the set X, denoted TX , consists of all transformations of X (i.e., all functions X → X) under the operation of composition. The symmetric group Sym(X) is the group of units of TX. It is well known [12, 58,94] that every automorphism of TX corresponds to conjugation by some fixed element of Sym(X), so that Aut(TX ) =∼ Sym(X); there are no special cases such as |X| = 6, here. However, TX has no anti-automorphisms (we assume |X ≥ 2|), as we now explain. Recall that Green’s relations are defined on a semigroup S by

xRy ⇔ xS1 = yS1, xL y ⇔ S1x = S1y, xJ y ⇔ S1xS1 = S1yS1, H = R ∩ L , D = R ∨ L = R ◦ L = L ◦ R.

Here, S1 denotes the monoid obtained from S by adjoining an identity element 1, if necessary. It is clear that if α : S → S is an anti-automorphism of the semigroup S, then for any x,y ∈ S,

xRy ⇔ α(x)L α(y), xL y ⇔ α(x)Rα(y), xK y ⇔ α(x)K α(y),

11 where K denotes any of H , D, J . In particular, α maps R-, L - and D-classes of S to L -, R- and D-classes of S (respectively), and if D is a D-class, then the number of L -classes in D is equal to the number of R-classes in α(D), with a dual statement holding also.

It is well known [49, 51] that for transformations f, g ∈TX ,

fRg ⇔ im(f) = im(g), fL g ⇔ ker(f) = ker(g), fDg ⇔ fJ g ⇔ rank(f) = rank(g), where, as usual, im(f), ker(f) and rank(f) denote the image, kernel and rank of f ∈ TX , and are defined by

im(f)= {f(x) : x ∈ X}, ker(f)= {(x,y) ∈ X × X : f(x)= f(y)}, rank(f)= |im(f)|.

Note that im(f) is a subset of X, ker(f) is an equivalence relation on X, and rank(X) is a cardinal (which may be anything from 1 to |X|). (Note also that it is common in the semigroup literature to write functions to the right of their arguments; with this convention, the roles of the image and kernel in characterising the R and L relations would be switched.) So the D = J -classes are given by

Dλ = {f ∈TX : rank(f)= λ} for each cardinal 1 ≤ λ ≤ |X|.

And the R-classes and L -classes contained in the D-class Dλ are indexed (respectively) by the sets

Rλ = {A ⊆ X : |A| = λ} and Lλ = {ε : ε is an equivalence on X with λ equivalence classes}, corresponding (respectively) to the collections of all images and kernels of the given rank λ. In particu- lar, |Rλ|≥ 2 and |Lλ|≥ 2 for any 1 ≤ λ ≤ |X| unless (a) λ = 1, where we have |R1| = |X| and |L1| = 1, or (b) λ = |X| < ℵ0, in which case |Rλ| = |Lλ| = 1. So if α : TX → TX was an anti-automorphism, then α(D1) would be a D-class of TX with just one R-class and |X| ≥ 2 L -classes. Since there is no such D-class of TX , it follows that there are no anti-automorphisms of TX . That is,

− ± Aut (TX )= ∅, Aut (TX ) = Aut(TX ) =∼ Sym(X), C (TX )= h∅i = {id}.

+ Note that, writing X↔ for the free commutative semigroup over X, as in Section 4, we have ∼ + ± ∼ ± + ∼ + Aut(TX ) = Aut(X↔) and Aut (TX ) = Aut (X↔) but C (TX ) =6 C (X↔).

So the C (S) invariant enables one to distinguish some semigroups that have isomorphic (signed) auto- groups but are not isomorphic. (Of course many other invariants could be used to distinguish + between TX and X↔.)

5.6 Symmetric and dual symmetric inverse semigroups

There are natural analogues of the full transformation semigroup: namely, the sym- ∗ metric inverse semigroup IX [67,69], and the dual symmetric inverse semigroup IX [40]. These consist, respectively, of all between of X and all bijections between quotients of X, under ∗ ∼ operations we will not need to describe here. It is well known [72] that for |X|≥ 3, Aut(IX ) = Sym(X), ∗ with all automorphisms acting as conjugation by elements of Sym(X) ⊆ IX. It is also certainly well known that Aut(IX ) =∼ Sym(X) for any X; though we are unaware of a reference for this, the argument ∼ ∗ used in [58] to show that Aut(TX ) = Sym(X) may easily be adapted to show this. Since IX and IX are both inverse monoids, the involution ι : f 7→ f −1 commutes with every automorphism, so it follows ∗ that if S is either IX or IX (with |X|≥ 3 in the latter case), then

± Aut(S) =∼ Sym(X) and C (S) = Aut (S) =∼ Z2 × Sym(X).

12 5.7 Finite diagram monoids

(Finite) partition algebras were introduced independently by Paul Martin [73] and Vaughan Jones [62] in the context of Potts models in statistical mechanics, and are twisted semigroup algebras of the partition monoids [98]. The partition monoid PX on the set X contains (isomorphic copies of) the full transformation semigroup TX as well as the symmetric and dual symmetric inverse monoids ∗ IX and IX [27, 28, 30], and many other diagram monoids [4, 5, 26, 39, 65, 72, 74, 76]. The elements of PX have natural diagrammatic representations, but we will not explicitly describe these (or the of PX ); for more details, we refer the reader to [6, 25–27, 30, 76] and references therein. The partition monoid is a canonical example of a regular ∗-semigroup [82]; there is a map ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ : PX → PX : f 7→ f satisfying (f ) = f, (fg) = g f and f = ff f (and f = f ff ) for ∗ all f, g ∈ PX . In particular, the map is an involution; this involution and another were studied in [6] in the context of (inherently) non-finitely based equational theories. The partition monoid PX contains the symmetric group Sym(X) as its group of units, and if f ∈ Sym(X), then f ∗ = f −1. It is known [76] that Aut(PX ) =∼ Sym(X) for finite X, with all automorphisms corresponding to conjugation −1 ∗ ∗ −1 ∗ by permutations. Since (fgf ) = f(g )f for all f ∈ Sym(X) and g ∈ PX , we see that the map commutes with each automorphism of (finite) PX , and so

± Aut(PX ) =∼ Sym(X) and C (PX ) = Aut (PX ) =∼ Z2 × Sym(X).

5.8 Full linear monoids

Let Mn(C) denote the monoid of all n×n matrices over the complex field C. We also write GL(n, C)= {A ∈ Mn(C) : det(A) 6= 0} for the of degree n over C, which is the group of units of Mn(C). For an invertible matrix A ∈ GL(n, C), denote by φA ∈ Aut(Mn(C)) the automorphism −1 determined by conjugation by A. That is, φA : Mn(C) → Mn(C) is defined by φA(X) = AXA for all X ∈ Mn(C). Putcha [87] showed that

Aut(Mn(C)) = {φA : A ∈ GL(n, C)}.

So Φ : GL(n, C) → Aut(Mn(C)) : A 7→ φA is a surjective , and its ker- nel, ker(Φ) = {A ∈ GL(n, C) : φA = id}, is precisely the centre Z(GL(n, C)) of GL(n, C); that is, ker(Φ) = Z(GL(n, C)) = {xIn : x ∈ C \ {0}}. In particular, Aut(Mn(C)) is isomorphic to the quotient GL(n, C)/Z(GL(n, C)), which is precisely the projective general linear group of degree n over C, and denoted PGL(n, C).

T Denote by τ : Mn(C) → Mn(C) : A 7→ A the transpose map. Since τ is an anti-automorphism (indeed, an involution), Lemma 2.1 tells us that every anti-automorphism of Mn(C) is of the form φAτ for some A ∈ GL(n, C); we will write ψA = φAτ for this anti-automorphism. So

± Aut (Mn(C)) = {φA : A ∈ GL(n, C)} ∪ {ψA : A ∈ GL(n, C)}.

Note that T −1 ψA(X)= AX A for all X ∈ Mn(C).

One easily checks that τφA = φ(AT)−1 τ = ψ(AT)−1 , and that the inverse mapping of ψA is ψAT . In particular,

C T C ψA ∈ I(Mn( )) ⇔ ψA = ψAT ⇔ φA = φAT ⇔ A = xA for some x ∈ \ {0}.

T So, for example, ψA ∈ I(Mn(C)) if A is symmetric (i.e., A = A ). It is well known [10] that any matrix A ∈ Mn(C) is the product A = A1A2 of two symmetric matrices A1, A2 ∈ Mn(C). In particular, if A ∈ GL(n, C), then the symmetric matrices A1, A2 also belong to GL(n, C), and we have

2 T −1 C φA = φA1 φA2 = φA1 τ φA2 = ψA1 ψ(A2 ) ∈ C (Mn( )),

13 T −1 −1 C C since (A2 ) = A2 is also symmetric. This shows that Aut(Mn( )) ⊆ C (Mn( )), and it follows that ± C (Mn(C)) = Aut (Mn(C)) = Aut(Mn(C)) ⋊ hτi =∼ PGL(n, C) ⋊Z2.

Note here that the semidirect product is not direct, in view of the rule τφA = φ(AT)−1 τ. We also note that, since any complex matrix of determinant 1 is a product of (at most four) involutory matrices [44], it quickly follows that Aut(Mn(C)) =∼ PGL(n, C) is itself involution-generated; that is, Aut(Mn(C)) = G (Mn(C)) = hJ(Mn(C))i.

5.9 Rectangular bands

A is a semigroup consisting entirely of idempotents. It is well known [14] that every band is a of rectangular bands. A rectangular band is a semigroup B = X × Y with multiplication (x1,y1)(x2,y2) = (x1,y2) for x1,x2 ∈ X and y1,y2 ∈ Y . In this section, we fix such a rectangular band B = X × Y , and we describe the group C (B).

The R-, L - and H -classes of B are precisely the sets

Rx = {x}× Y, Ly = X × {y},Hxy = Rx ∩ Ly = {(x,y)} for each x ∈ X and y ∈ Y .

Since any anti-automorphism of B maps R-classes to L -classes (and vice versa), Aut−(B) = ∅ if |X|= 6 |Y |, in which case C (B) = {idB}. So we assume from now on that |X| = |Y |; in fact, without loss of generality, we may assume that X = Y . So B = X × X is a square band.

For σ, τ ∈ Sym(X), we define mappings

γσ,τ : B → B : (x,y) 7→ (σ(x),τ(y)) and δσ,τ : B → B : (x,y) 7→ (σ(y),τ(x)).

Note that γid,id = idB. It is easily seen (and well known in the case of automorphisms) that

− Aut(B)= {γσ,τ : σ, τ ∈ Sym(X)} =∼ Sym(X) × Sym(X) and Aut (B)= {δσ,τ : σ, τ ∈ Sym(X)}.

(More generally, Aut(X × Y ) =∼ Sym(X) × Sym(Y ).)

Put ι = δid,id, so ι(x,y) = (y,x) for all x,y ∈ X. Then clearly ι ∈ I(S), so Proposition 3.2 gives

± Aut (B) = Aut(B) ⋊ hιi =∼ (Sym(X) × Sym(X)) ⋊Z2 = Sym(X) ≀ Z2.

Here G ≀ Z2 = (G × G) ⋊Z2 denotes the wreath product in which Z2 =∼ Sym(2) acts on G × G by permuting the coordinates. Specifically, the action of hιi =∼ Z2 on Aut(B) =∼ Sym(X) × Sym(X) is given by ιγσ,τ ι = γτ,σ for all σ, τ ∈ Sym(X). We now describe the group C (B)= hI(B)i. One easily checks that products in Aut±(B) are given by

γσ1,τ1 γσ2,τ2 = γσ1σ2,τ1τ2 , δσ1,τ1 δσ2,τ2 = γσ1τ2,τ1σ2 , γσ1,τ1 δσ2,τ2 = δσ1σ2,τ1τ2 , δσ1,τ1 γσ2,τ2 = δσ1τ2,τ1σ2 .

−1 In particular, δσ,τ ∈ I(B) is an involution if and only if τ = σ , so

I(B)= {δσ,σ−1 : σ ∈ Sym(X)}.

By Proposition 3.2, C (B) = (C (B) ∩ Aut(B)) ⋊ hιi, so we concentrate on the group C (B) ∩ Aut(B) which, for convenience, we will denote by G. So G consists of all even-length products of elements from I(B). Note that if σ, τ ∈ Sym(X) are arbitrary, then

δσ,σ−1 δτ −1,τ = γστ,σ−1τ −1 = γσ,σ−1 γτ,τ −1 , and that γσ,σ−1 = δσ,σ−1 ι ∈ G (and similarly γτ,τ −1 ∈ G). So in fact, G is generated by the set

{γσ,σ−1 : σ ∈ Sym(X)}.

14 ∼ It quickly follows that G = KSym(X), in the notation of Lemma 2.4. By the above discussion, and the ± observations in Example 2.7, we see that C (B) = Aut (B) =∼ Sym(X) ≀ Z2 for infinite X, and

C (B)= {γσ,τ , δσ,τ : σ, τ ∈ Sym(X), στ ∈ Alt(X)} =∼ (Alt(X) × Alt(X)) ⋊Z2 ⋊Z2 for finite X. Let X be finite and put 

K = KSym(X) = {(σ, τ) ∈ Sym(X) × Sym(X) : στ ∈ Alt(X)}. So, as noted in Example 2.7, Alt(X) × Alt(X) ≤ K ≤ Sym(X) × Sym(X). In this way, we see that

Alt(X) ≀ Z2 ≤ K ⋊Z2 ≤ Sym(X) ≀ Z2, so C (B) =∼ K ⋊Z2 is a sub-wreath product; such products are also called cascade products [33].

5.10 The doubled semigroup construction

We have seen examples of semigroups with no anti-automorphisms; these include the full transformation semigroups TX (Section 5.5) and non-square rectangular bands (Section 5.9). In this section, we provide − a construction that allows us to extend such a semigroup S with Aut (S)= ∅ to a semigroup DS with − Aut (DS) 6= ∅; indeed, I(DS ) 6= ∅ and the involutions of DS are in one-one correspondence with the automorphisms of S.

With the above discussion in mind, let S be an arbitrary non- with Aut−(S) = ∅. − ± (The following construction works also in the case that Aut (S) 6= ∅, but the calculations of Aut (DS) ∗ and C (DS) are more involved; the authors can be contacted for details.) Let S = {s : s ∈ S} be a set disjoint from S, and in one-one correspondence with S via the map s 7→ s∗. Define a product on S∗ by s∗t∗ = (ts)∗ for each s,t ∈ S. So S∗ is the dual semigroup of S. Now define a new semigroup ∗ ∗ DS = S ∪ S ∪ {0}, where 0 is a new symbol that does not belong to S (or S ). We call DS the doubled semigroup obtained from S. Define a product · on DS, for x,y ∈ DS, by

xy if x,y ∈ S or x,y ∈ S∗ x · y = (0 otherwise. ∗ So DS contains both S and S as subsemigroups. Note that ι : DS → DS defined by ι(0) = 0, ι(s)= s∗, ι(s∗)= s for all s ∈ S

− is an involution of DS. In particular, I(DS ) (and hence Aut (DS)) is non-empty.

∗ ∗ Lemma 5.1. (i) If γ ∈ Aut(DS), then γ(S)= S and γ(S )= S . − ∗ ∗ (ii) If δ ∈ Aut (DS), then δ(S)= S and δ(S )= S.

∗ ∗ Proof. Since ι(S)= S and ι(S )= S, it suffices (by Lemma 2.1) to prove (i), so suppose γ ∈ Aut(DS). First note that if there exists s,t ∈ S such that γ(s) ∈ S and γ(t) ∈ S∗, then γ(0) = 0 = γ(s)γ(t) = γ(st), with st 6= 0, contradicting the injectivity of γ. It follows that either (a) γ(S) ⊆ S or (b) γ(S) ⊆ S∗. A similar argument shows that (a)∗ γ(S∗) ⊆ S or (b)∗ γ(S∗) ⊆ S∗. Since γ is surjective, we cannot have both (a) and (a)∗, and neither can we have both (b) and (b)∗. Next suppose (a)∗ and (b) hold. Since γ is surjective, we would have γ(S)= S∗ (and γ(S∗)= S). But then the restriction of γ to S would yield an isomorphism S → S∗, and composing this with the anti-isomorphism S∗ → S : s∗ 7→ s would yield an anti-automorphism of S, contradicting the assumption that Aut−(S) = ∅. It follows that (a) and (b)∗ hold. Again, the surjectivity of γ shows that γ(S)= S and γ(S∗)= S∗. ✷

For automorphisms α, β ∈ Aut(S), we define maps γα,β, δα,β : DS → DS by

∗ ∗ γα,β(0) = 0, γα,β(s) = α(s), γα,β(s ) = β(s) for all s ∈ S, ∗ ∗ δα,β(0) = 0, δα,β(s) = β(s) , δα,β(s ) = α(s) for all s ∈ S.

15 − It is easy to check that γα,β ∈ Aut(DS ) and δα,β ∈ Aut (DS) for all α, β ∈ Aut(S). It then follows quickly from Lemma 5.1 that

− Aut(DS )= {γα,β : α, β ∈ Aut(S)} and Aut (DS )= {δα,β : α, β ∈ Aut(S)}.

It is easily checked that for any α1, α2, β1, β2 ∈ Aut(S),

γα1,β1 γα2,β2 = γα1α2,β1β2 , δα1,β1 δα2,β2 = γα1β2,β1α2 , γα1,β1 δα2,β2 = δα1α2,β1β2 , δα1,β1 γα2,β2 = δα1β2,β1α2 .

Since also ιγα,βι = γβ,α for all α, β ∈ Aut(S), it follows that

± Aut(DS) =∼ Aut(S) × Aut(S) and Aut (DS ) = Aut(DS ) ⋊ hιi =∼ Aut(S) ≀ Z2.

In a similar fashion to Section 5.9, we have

I(DS )= {δα,α−1 : α ∈ Aut(S)}, and it again quickly follows that ∼ ∼ C (DS) ∩ Aut(DS) = KAut(S) and C (DS) = KAut(S) ⋊Z2. ∼ Again, we note that [Aut(S), Aut(S)] ≀ Z2 ≤ KAut(S) ⋊Z2 ≤ Aut(S) ≀ Z2, so that C (DS ) = KAut(S) ⋊Z2 is again a cascade product [33].

Example 5.2. Consider the case in which S = TX is the full transformation semigroup on the set X. − We saw in Section 5.5 that Aut (TX )= ∅ and Aut(TX ) =∼ Sym(X). It then follows that ∼ ± ∼ Z ∼ ⋊Z Aut(DTX ) = Sym(X) × Sym(X), Aut (DTX ) = Sym(X) ≀ 2, C (DTX ) = KSym(X) 2.

The group KSym(X) is described in Example 2.7. Interestingly, by consulting Section 5.9, we see that ∼ A(DTX ) = A(B), where B = X × X is a square band and A denotes any of the operators Aut, Aut±, C . We obtain the same isomorphisms in the case that S = X is a left-zero semigroup (in which case, S∗ = X∗ is a right zero semigroup).

5.11 Summary

Table 1 summarises the calculations of C (S) for the various semigroups S we have considered in Sections 4 and 5.

Semigroup S Aut(S) Aut±(S) C (S) Section + Free semigroup, X Sym(X) Z2 × Sym(X) Z2 × Sym(X) 4 + Free commutative semigroup, X↔ Sym(X) Sym(X) Sym(X) 4 Graph semigroup, SΓ (Γ 6= KX ) Aut(Γ) Z2 × Aut(Γ) Z2 × C (Γ) 4 Free group, FX , |X| < ℵ0 Aut(FX ) Z2 × Aut(FX ) Z2 × Aut(FX ) 5.1 Free abelian group, Zn GL(n, Z) GL(n, Z) GL(n, Z) 5.2 Z U U ZR(n) Finite cyclic group, n n n 2 5.3 Symmetric group, Sym(X), |X|= 6 1, 2, 6 Sym(X) Z2 × Sym(X) Z2 × Sym(X) 5.4 Symmetric group, Sym(X), |X| = 6 Sym(X) ⋊Z2 Z2 × (Sym(X) ⋊Z2) Z2 × (Sym(X) ⋊Z2) 5.4 Full transformation semigroup, TX Sym(X) Sym(X) {id} 5.5 Symmetric inverse monoid, IX Sym(X) Z2 × Sym(X) Z2 × Sym(X) 5.6 ∗ Z Z Dual symmetric inverse monoid, IX , |X|≥ 3 Sym(X) 2 × Sym(X) 2 × Sym(X) 5.6 Partition monoid, PX , |X| < ℵ0 Sym(X) Z2 × Sym(X) Z2 × Sym(X) 5.7 Full linear monoid, Mn(C) PGL(n, C) PGL(n, C) ⋊Z2 PGL(n, C) ⋊Z2 5.8 Rectangular band X × Y , |X|= 6 |Y | Sym(X) × Sym(Y ) Sym(X) × Sym(Y ) {id} 5.9 Square band X × X, |X|≥ℵ0 Sym(X) × Sym(X) Sym(X) ≀ Z2 Sym(X) ≀ Z2 5.9 Square band X × X, |X| < ℵ0 Sym(X) × Sym(X) Sym(X) ≀ Z2 ((Alt(X) × Alt(X)) ⋊Z2) ⋊Z2 5.9 − Doubled semigroup DS, Aut (S)= ∅ Aut(S) × Aut(S) Aut(S) ≀ Z2 KAut(S) ⋊Z2 5.10 Z Z Doubled semigroup DTX , |X|≥ℵ0 Sym(X) × Sym(X) Sym(X) ≀ 2 Sym(X) ≀ 2 5.10 Z ⋊Z ⋊Z Doubled semigroup DTX , |X| < ℵ0 Sym(X) × Sym(X) Sym(X) ≀ 2 ((Alt(X) × Alt(X)) 2) 2 5.10

Table 1: Semigroups S, along with Aut(S), Aut±(S) and C (S). See the specified sections for further information.

16 6 Open problems

We conclude the article with a short list of open problems that we believe are worth investigating.

Problem 1. Classify the (involution-generated) groups G for which G =∼ C (S) for some semigroup S with a proper involution.

Problem 2. Classify the (involution-generated) groups G for which G =∼ C (H) for some group H.

Problem 3. Classify the semigroups S for which C (S) = Aut±(S).

Problem 4. Describe the group C (S) in the case that S is one of the following: a band; a completely 0-simple semigroup; a semigroup of matrices over any field; a finite abelian group; a free group or free abelian group of infinite rank; a graph group; an infinite partition monoid.

References

[1] [2] J. Ara´ujo, P. V. B¨unau, J. D. Mitchell, and M. Neunh¨offer. Computing automorphisms of semigroups. J. Symbolic Comput., 45(3):373–392, 2010. [3] William Arveson. An invitation to C∗-algebras. Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 39. [4] Karl Auinger. Krohn-Rhodes complexity of Brauer type semigroups. Port. Math., 69(4):341–360, 2012. [5] Karl Auinger. Pseudovarieties generated by Brauer type monoids. Forum Math., 26(1):1–24, 2014. [6] Karl Auinger, Igor Dolinka, and Mikhail V. Volkov. Equational theories of semigroups with involution. J. Algebra, 369:203–225, 2012. [7] Karl Auinger, Igor Dolinka, and Mikhail V. Volkov. Matrix identities involving multiplication and transposition. J. Eur. Math. Soc. (JEMS), 14(3):937–969, 2012. [8] Martin Bacovsk´y. MI-groups: New approach. In Anne Laurent, Olivier Strauss, Bernadette Bouchon-Meunier, and Ronald R. Yager, editors, Information Processing and Management of Uncertainty in Knowledge-Based Sys- tems, volume 443 of Communications in Computer and Information Science, pages 274–283. Springer International Publishing, 2014. [9] Sangeeta Bhatia, Attila Egri-Nagy, and Andrew R. Francis. Algebraic double cut and join. Journal of mathematical biology, pages 1–30, 2014. [10] A. J. Bosch. The factorization of a square matrix into two symmetric matrices. Amer. Math. Monthly, 93(6):462–464, 1986. [11] R. W. Carter. Conjugacy classes in the Weyl group. Compositio Math., 25:1–59, 1972. [12] Suchat Chantip and G. R. Wood. Automorphisms of the semigroup of all onto mappings of a set. Bull. Austral. Math. Soc., 14(3):399–405, 1976. [13] Laura Ciobanu, Volker Diekert, and Murray Elder. Solution sets for equations over free groups are EDT0L languages. Preprint, 2015, arXiv:1502.03426. [14] A. H. Clifford. Bands of semigroups. Proc. Amer. Math. Soc., 5:499–504, 1954. [15] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson. Atlas of finite groups. Oxford University Press, Eynsham, 1985. Maximal and ordinary characters for simple groups, With computational assistance from J. G. Thackray. [16] Siniˇsa Crvenkovi´cand Igor Dolinka. Varieties of involution semigroups and involution : a survey. Bull. Soc. Math. Banja Luka, 9:7–47 (2004), 2002. [17] S. D˘asc˘alescu, C. N˘ast˘asescu, and M. N˘ast˘asescu. Strongly involutory functors. Comm. Algebra, 37(5):1677–1689, 2009. [18] J. de Groot. Groups represented by homeomorphism groups. Math. Ann., 138:80–102, 1959. [19] H. de Vries and A. B. de Miranda. Groups with a small number of automorphisms. Math. Z, 68:450–464, 1958. [20] D. Z.ˇ Djokovi´c. Product of two involutions. Arch. Math. (Basel), 18:582–584, 1967. [21] Igor Dolinka. All varieties of normal bands with involution. Period. Math. Hungar., 40(2):109–122, 2000. [22] Igor Dolinka. Remarks on varieties of involution bands. Comm. Algebra, 28(6):2837–2852, 2000. [23] Igor Dolinka. On the of varieties of involution semigroups. Semigroup Forum, 62(3):438–459, 2001. [24] Igor Dolinka. On identities of finite involution semigroups. Semigroup Forum, 80(1):105–120, 2010.

17 [25] Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Lough- lin. Enumeration of idempotents in diagram semigroups and algebras. J. Combin. Theory Ser. A, 131:119–152, 2015. [26] J. East and R.D. Gray. Diagram monoids and Graham–Houghton graphs: idempotents and generating sets of ideals. Preprint, 2015, arXiv:1404.2359. [27] James East. Generators and relations for partition monoids and algebras. J. Algebra, 339:1–26, 2011. [28] James East. On the singular part of the partition monoid. Internat. J. Algebra Comput., 21(1-2):147–178, 2011. [29] James East. A symmetrical presentation for the singular part of the symmetric inverse monoid. Algebra Universalis, (to appear). [30] James East and D. G. FitzGerald. The semigroup generated by the idempotents of a partition monoid. J. Algebra, 372:108–133, 2012. [31] Attila Egri-Nagy, Andrew R. Francis, and Volker Gebhardt. Bacterial genomics and computational : The biogappackage for gap. In Hoon Hong and Chee Yap, editors, Mathematical Software ICMS 2014, volume 8592 of Lecture Notes in , pages 67–74. Springer Berlin Heidelberg, 2014. [32] Attila Egri-Nagy, Volker Gebhardt, Mark M. Tanaka, and Andrew R. Francis. Group-theoretic models of the inversion process in bacterial genomes. Journal of Mathematical Biology, 69(1):243–265, 2014. [33] Attila Egri-Nagy and Chrystopher L. Nehaniv. Cascade product of permutation groups. Preprint, 2014, arXiv:1303.0091. [34] Brent Everitt and John Fountain. Partial , reflection monoids and Coxeter groups. Adv. Math., 223(5):1782–1814, 2010. [35] Brent Everitt and John Fountain. Partial mirror symmetry, lattice presentations and algebraic monoids. Proc. Lond. Math. Soc. (3), 107(2):414–450, 2013. [36] R. Exel. Partial dynamical systems, Fell bundles and applications. Licensed under a Creative Commons Attribution- ShareAlike 4.0 International License. Available online from mtm.ufsc.br/~exel/publications. 2014. ∗ [37] Ruy Exel. Invertibility in C -algebras. In Operator theory, operator algebras and applications, volume 242 of Oper. Theory Adv. Appl., pages 173–183. Birkh¨auser/Springer, Basel, 2014. [38] David C. Fisher. The number of words of length n in a graph monoid. Amer. Math. Monthly, 96(7):610–614, 1989. [39] D. G. FitzGerald and Kwok Wai Lau. On the partition monoid and some related semigroups. Bull. Aust. Math. Soc., 83(2):273–288, 2011. [40] D. G. FitzGerald and Jonathan Leech. Dual symmetric inverse monoids and . J. Austral. Math. Soc. Ser. A, 64(3):345–367, 1998. [41] Robert Frucht. Herstellung von graphen mit vorgegebener abstrakter gruppe. Compos. Math., 6:239–250, 1938. [42] Robert Frucht. Graphs of degree three with a given abstract group. Canadian J. Math., 1:365–378, 1949. [43] J. J. Graham and G. I. Lehrer. Cellular algebras. Invent. Math., 123(1):1–34, 1996. [44] W. H. Gustafson, P. R. Halmos, and H. Radjavi. Products of involutions. and Appl., 13(1/2):157–162, 1976. Collection of articles dedicated to Olga Taussky Todd. [45] William H. Gustafson. On products of involutions. In Paul Halmos, pages 237–255. Springer, New York, 1991. [46] Jurriaan Hage and Tero Harju. On involutive anti-automorphisms of finite abelian groups. Department of Information and Computing Sciences, Utrecht University, Technical Report UU-CS-2007-053, Available at http://www.cs.uu.nl/research/techreps/repo/CS-2007/2007-053.pdf. [47] Henry E. Heatherly, Enoch K. S. Lee, and Richard Wiegandt. Involutions on universal algebras. In Nearrings, nearfields and K-loops (Hamburg, 1995), volume 426 of Math. Appl., pages 269–282. Kluwer Acad. Publ., Dordrecht, 1997. [48] Zdenˇek Hedrl´ın and Joachim Lambek. How comprehensive is the of semigroups? J. Algebra, 11:195–212, 1969. [49] Peter M. Higgins. Techniques of semigroup theory. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. [50] Peter M. Higgins. Permutations of a semigroup that map to inverses. Semigroup Forum, 89(1):169–182, 2014. [51] John 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. [52] James E. Humphreys. Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathe- matics. Cambridge University Press, Cambridge, 1990. [53] Teruo Imaoka. On fundamental regular ∗-semigroups. Mem. Fac. Sci. Shimane Univ., 14:19–23, 1980. [54] Teruo Imaoka. ∗-congruences on regular ∗-semigroups. Semigroup Forum, 23(4):321–326, 1981. [55] Teruo Imaoka. Prehomomorphisms on regular ∗-semigroups. Mem. Fac. Sci. Shimane Univ., 15:23–27, 1981. ∗ [56] Teruo 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. [57] Hiroyuki Ishibashi. Involutary expressions for elements in GLn(Z) and SLn(Z). Linear Algebra Appl., 219:165–177, 1995. [58] Nobuko Iwahori and Hirosi Nagao. On the automorphism group of the full transformation semigroups. Proc. Japan Acad., 48:639–640, 1972. [59] Herbert Izbicki. Unendliche Graphen endlichen Grades mit vorgegebenen Eigenschaften. Monatsh. Math., 63:298– 301, 1959.

18 [60] Peter R. Jones. A common framework for restriction semigroups and regular ∗-semigroups. J. Pure Appl. Algebra, 216(3):618–632, 2012. [61] Peter R. Jones. Varieties of P -restriction semigroups. Comm. Algebra, 42(4):1811–1834, 2014. [62] 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. [63] William M. Kantor. Those nasty Baer involutions. In Proceedings of the International Conference Projective Planes (Washington State Univ., Pullman, Wash., 1973), pages 145–155. Washington State Univ. Press, Pullman, Wash., 1973. [64] Martin Kassabov. Generating mapping class groups by involutions. Preprint, 2003, arXiv:0311455. [65] Kwok Wai Lau and D. G. FitzGerald. Ideal structure of the Kauffman and related monoids. Comm. Algebra, 34(7):2617–2629, 2006. [66] Michael R. Laurence. A generating set for the automorphism group of a graph group. J. London Math. Soc. (2), 52(2):318–334, 1995. [67] Mark V. Lawson. Inverse semigroups. World Scientific Publishing Co., Inc., River Edge, NJ, 1998. The theory of partial symmetries. [68] Christian Liedtke. Natural central extensions of groups. Groups Geom. Dyn., 2(2):245–261, 2008. [69] Stephen Lipscomb. Symmetric inverse semigroups, volume 46 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1996. [70] George Lusztig and David A. Vogan, Jr. Hecke algebras and involutions in Weyl groups. Bull. Inst. Math. Acad. Sin. (N.S.), 7(3):323–354, 2012. [71] George Lusztig and David A. Vogan, Jr. Hecke algebras and involutions in Coxeter groups. Preprint, 2014, arXiv:1312.3237. ∗ [72] Victor Maltcev. On a new approach to the dual symmetric inverse monoid IX . Internat. J. Algebra Comput., 17(3):567–591, 2007. [73] Paul Martin. Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J. Knot Theory Ramifications, 3(1):51–82, 1994. [74] Paul Martin and Volodymyr Mazorchuk. On the representation theory of partial Brauer algebras. Q. J. Math., 65(1):225–247, 2014. [75] Helmut M¨aurer. Zur Automorphismengruppe der symmetrischen Gruppen. Mitt. Math. Ges. Hamburg, 11(2):265– 266, 1983. [76] Volodymyr Mazorchuk. Endomorphisms of Bn, PBn, and Cn. Comm. Algebra, 30(7):3489–3513, 2002. [77] Jon McCammond. The exceptional symmetry. Preprint, 2014, arXiv:1412.1855. [78] Naoyuki Monden. Generating the mapping class group of a punctured by involutions. Tokyo J. Math., 34(2):303–312, 2011. [79] W. D. Munn. Free inverse semigroups. Proc. London Math. Soc. (3), 29:385–404, 1974. [80] Trygve Nagell. Introduction to Number Theory. John Wiley & Sons, Inc., New York; Almqvist & Wiksell, Stockholm, 1951. [81] Jakob Nielsen. Die Isomorphismengruppe der freien Gruppen. Math. Ann., 91(3-4):169–209, 1924. [82] T. E. Nordahl and H. E. Scheiblich. Regular ∗-semigroups. Semigroup Forum, 16(3):369–377, 1978. [83] Oystein Ore. Some remarks on commutators. Proc. Amer. Math. Soc., 2:307–314, 1951. [84] Dmitri I. Panyushev. Commuting involutions and degenerations of isotropy representations. Transform. Groups, 18(2):507–537, 2013. [85] Mario Petrich. Certain varieties of completely regular ∗-semigroups. Boll. Un. Mat. Ital. B (6), 4(2):343–370, 1985. [86] Libor Pol´ak. A solution of the word problem for free ∗-regular semigroups. J. Pure Appl. Algebra, 157(1):107–114, 2001. [87] Mohan S. Putcha. On the automorphism group of a linear algebraic monoid. Proc. Amer. Math. Soc., 88(2):224–226, 1983. ∗ [88] N. R. Reilly. A class of regular -semigroups. Semigroup Forum, 18(4):385–386, 1979. [89] Gert Sabidussi. Graphs with given group and given graph-theoretical properties. Canad. J. Math., 9:515–525, 1957. [90] H. E. Scheiblich. Free inverse semigroups. Proc. Amer. Math. Soc., 38:1–7, 1973. [91] H. E. Scheiblich. The free elementary ∗-. In Semigroups (Proc. Conf., Monash Univ., Clayton, 1979), pages 191–206. Academic Press, New York-London, 1980. [92] H. E. Scheiblich. semigroups with involution. Rocky Mountain J. Math., 12(2):205–211, 1982. [93] H. E. Scheiblich. Projective and injective bands with involution. J. Algebra, 109(2):281–291, 1987. [94] J. Schreier. Uber¨ abbildungen einer abstrakten menge auf ihre teilmengen. Fundamenta Mathematicae, 28(1):261– 264, 1937. [95] J. Schreier and Stanis law Ulam. Uber¨ die automorphismen der permutationsgruppe der nat¨urlichen zahlenfolge. Fundamenta Mathematicae, 28(1):258–260, 1937. [96] Herman Servatius. Automorphisms of graph groups. J. Algebra, 126(1):34–60, 1989. ∗ [97] N. E. Wegge-Olsen. K-theory and C -algebras. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. A friendly approach. [98] Stewart Wilcox. Cellularity of diagram algebras as twisted semigroup algebras. J. Algebra, 309(1):10–31, 2007. [99] S. K. Winker, L. Wos, and E. L. Lusk. Semigroups, antiautomorphisms, and involutions: a computer solution to an open problem. I. Math. Comp., 37(156):533–545, 1981. [100] Mar´ıa J. Wonenburger. Transformations which are products of two involutions. J. Math. Mech., 16:327–338, 1966. [101] Miyuki Yamada. Note on the construction of regular ∗-semigroups. Mem. Fac. Sci. Shimane Univ., 15:17–22, 1981.

19