The algebra of rewriting for presentations of inverse monoids
aa r X i v : . [ m a t h . G R ] A p r THE ALGEBRA OF REWRITING FOR PRESENTATIONS OF INVERSEMONOIDS
N.D. GILBERT AND E.A. MCDOUGALL
Department of Mathematics and the Maxwell Institute for the Mathematical Sciences,Heriot-Watt University, Edinburgh, EH14 4AS A BSTRACT . We describe a formalism, using groupoids, for the study of rewriting forpresentations of inverse monoids, that is based on the Squier complex construction formonoid presentations. We introduce the class of pseudoregular groupoids, an exampleof which now arises as the fundamental groupoid of our version of the Squier complex.A further key ingredient is the factorisation of the presentation map from a free inversemonoid as the composition of an idempotent pure map and an idempotent separating map.The relation module of a presentation is then defined as the abelianised kernel of thisidempotent separating map. We then use the properties of idempotent separating mapsto derive a free presentation of the relation module. The construction of its kernel - themodule of identities - uses further facts about pseudoregular groupoids. I NTRODUCTION
Inverse semigroups (and inverse monoids) comprise a class of algebraic structures that sitnaturally between the class of semigroups and the class of groups, and are the natural can-didates for semigroups that are structurally closest to groups. However, inverse semigrouppresentations do not sit quite so naturally between semigroup presentations and group pre-sentations, but have particular features that set them apart. For example, a finitely generatedfree inverse semigroup is not finitely presented as a semigroup [24], does not have a regularlanguage of normal forms [8], and no free inverse monoid has context-free word problem[1].In this paper we consider presentations of inverse monoids as rewriting systems, and at-tempt to replicate the formalism for describing rewriting in monoid presentations due toSquier [25, 26], and for group presentations due to Cremanns-Otto [6] and Pride [22].Given a monoid presentation P with generating set A , Squier associates to P a graph Γ that has vertex set A ∗ (the free monoid on A ) and, for all p, q ∈ A ∗ , has an edge from puq to pvq whenever u = v is a relation in P . A path in Γ therefore corresponds to a chain ofequivalences betwen words in A ∗ as consequences of the relations in P , and a homotopyrelation is imposed to identify paths corresponding to such equivalences that are naturallyconsidered to be essentially the same. If this homotopy relation is finitely generated, then E-mail address : [email protected] (corresponding author).2010
Mathematics Subject Classification.
Primary: 20M18, Secondary: 20L05, 20M50.
Key words and phrases. inverse monoid, presentation, groupoid, crossed module. P is said to have finite derivation type . For monoid presentations of groups, a theorem ofSquier [26, Theorem 4.3] shows that if one finite presentation of G has finite derivationtype then all finite presentations of G do. The main result of [6] is that finite derivationtype (for groups) is equivalent to the homological finiteness property FP .An important component of the treatment of groups (given by monoid presentations) in [6]is the way in which free reductions are handled within the formalism. An approach basedon the categorical algebra of monoidal groupoids and crossed modules, and refashioningthe results of [22], was given in [9]. This approach is refined and extended in [11]. Inany similar approach to presentations of inverse monoids, we encounter the problem ofhandling the Wagner congruence (see [27], for example), which defines the free inversemonoid as a quotient of a free monoid, and as mentioned above, is not finitely generated.To get around this problem , given a presentation P = [ X : R ] of an inverse monoid M ,we define a –complex as in [14, 21] whose edges encode the applications of relations,and whose –cells impose an appropriate homotopy relation, but we take as vertex set aninverse monoid T constructed canonically from P . The presentation map FIM( X ) → M from the free inverse monoid on X to M factors through T , which has M as an idempotentseparating image. We then work with the fundamental groupoid of this –complex: theuse of groupoids in this general setting originates with the work of Kilibarda [14]. Asa groupoid whose set of identities is an inverse monoid, our fundamental groupoid is anexample of a pseudoregular groupoid, whose properties are considered in section 2. Wethen aim to connect the structure of the fundamental groupoid with the relation module of P : for group presentations the FP condition is equivalent to finite presentation of therelation module.We define the relation module of P in section 3. We take a more direct approach than inearlier work of the first author [10], since the relation module can now be naturally definedin terms of the map T → M , and as in [10] we show that the relation module is isomorphicto the first homology of the Sch¨utzenberger graph of ( M, X ) . In section 4 we establish theconnection between the relation module and the fundamental groupoid of our –complex.We use an intermediate construction of a free crossed module of groupoids, and derive afree presentation of the relation module as an M –module.1. B ACKGROUND NOTIONS AND NOTATION
Our basic reference for the theory of inverse semigroups is Lawson’s book [16]. Aspectsof the theories of groups and inverse semigroups are considered by side-by-side in [19].We shall also make use of other algebraic constructions that may be less familiar, and wegive brief introductions here.1.1.
Groupoids. A groupoid G is a small category in which every morphism is invertible.We consider a groupoid as an algebraic structure (as in [12, 16]) whose elements are itsmorphisms, with a partial associative partial binary operation given by composition ofmorphisms. The set of vertices of G is denoted V ( G ) , and for each vertex x ∈ V ( G ) thereexists an identity morphism x . An element g ∈ G has domain g d and range g r in V ( G ) ,with gg − = 1 g d and g − g = 1 g r . For e ∈ V ( G ) the star of e in G is the set star e ( G ) = { g ∈ G : g d = e } , and the local group at e is the set G ( e ) = { g ∈ G : g d = e = g r } . EWRITING FOR PRESENTATIONS OF INVERSE MONOIDS 3
Example 1.1.
Let X be any set and let ρ ⊆ X × X be an equivalence relation on X .Then ρ is a groupoid with vertex set X , and with partial composition ( a, b )( c, d ) = ( a, d ) if b = c . If ρ = X × X we obtain the simplicial groupoid on X . Example 1.2.
Let X be a topological space and A a subspace of X . Then the set offixed-end-point homotopy classes of paths in X with end-points in A is a groupoid, the fundamental groupoid π ( X, A ) . We shall make use of the fundamental groupoid π ( X, A ) of a –complex X , with A its –skeleton, in section 4. Example 1.3.
An inverse semigroup S may be considered as a groupoid ~S , with V ( ~S ) equal to the set of idempotents E ( S ) of S . The groupoid composition ◦ on ~S is the re-stricted product on S : the composition s ◦ t is defined if and only if s − s = tt − , and then s ◦ t = st ∈ S . This point of view is an important theme in [16].1.2. Clifford Semigroups.
Clifford semigroups constitute a class of inverse semigroupsthat will be of importance in the description of relation modules in section 3.Let ( E, ) be a meet semilattice, and let { G e : e ∈ E } be a family of groups indexed bythe elements of E . For each pair e, f ∈ E with e > f , let φ ef : G e → G f be a grouphomomorphism, and suppose that the following two axioms hold: • φ ee is the identity homomorphism on G e , • if e > f > g then φ ef φ fg = φ eg . The collection ( G e , φ ef ) = ( { G e : e ∈ E } , { φ ef : e, f ∈ E, f e } ) is a presheaf of groups over E and the group operations on the G e make the disjoint union G = F e ∈ E G e into an inverse semigroup, called a Clifford semigroup over E , with binaryoperation x ∗ y = ( xφ eef )( yφ fef ) ∈ G ef , where x ∈ G e and y ∈ G f .Our description of relation modules in section 3 also depends on the factorization of aninverse semigroup homomorphism from a free inverse monoid as a composition of anidempotent pure map and an idempotent separating map. We recall the definitions of thesetypes of map here: Definition 1.1. (a) A congruence ρ on an inverse semigroup T is said to be idempotent pure if a ∈ T and a ρ e for some e ∈ E ( T ) imply that a ∈ E ( T ) .(b) A congruence ρ on an inverse semigroup T is said to be idempotent separating if e, f ∈ E ( T ) and e ρ f imply that e = f .Any inverse semigroup homomorphism φ : T → S induces a congruence χ φ on T by a χ φ b ⇐⇒ aφ = bφ . We say that φ is idempotent pure (respectively, idempotent separating) if χ φ has this prop-erty. The kernel of an inverse semigroup homomorphism φ : T → S is the preimage of E ( S ) : ker φ = { a ∈ T : aφ ∈ E ( S ) } . REWRITING FOR PRESENTATIONS OF INVERSE MONOIDS
We recall that any inverse semigroup T has a maximum group image b T and that T is E –unitary if the quotient map σ T : T → b T is idempotent pure. Free inverse monoids are E –unitary. See [16, section 2.4] for more properties of E –unitary inverse semigroups.The connection that we need between Clifford semigroups and idempotent separating mapsis given by the following result (see [16, Lemma 5.2.2]). Proposition 1.4.
If a homomorphism φ : T → S of inverse semigroups is idempotentseparating then its kernel is a Clifford semigroup over E ( T ) . Sch ¨utzenberger graphs.
We shall use left
Sch¨utzenberger graphs in this paper. Let S be an inverse semigroup generated by a set X . There exists a presentation map θ :FIS( X ) → S from the free inverse semigroup on X to S . The (left) Sch¨utzenberger graph Sch L ( S, X ) has vertex set S , and for x ∈ X and s ∈ S , an x –labelled edge from s to ( xθ ) s whenever ( x − x ) θ > ss − . The connected component Sch L ( S, X, e ) containingthe idempotent e is the full subgraph on the vertex set L e , the L –class of e in S . Someexamples of Sch¨utzenberger graphs may be found in section 3.1.1.4. Modules for inverse semigroups.
Modules for inverse semigroups were first definedby Lausch [15].
Definition 1.2.
Let S be an inverse semigroup with semilattice of idempotents E ( S ) .Consider a Clifford semigroup A = ( A e , α ef ) (see section 1.2), in which each A e is anadditively written abelian group with identity e . The disjoint union A = F e ∈ E ( S ) A e is acommutative inverse semigroup under the operation a ⊕ b = aα eef + bα fef for a ∈ A e and b ∈ A f . Then A is an S –module [15, section 2] if there exists a map A × S → A , written ( a, s ) a ⊳ s , such that(i) ( a ⊕ b ) ⊳ s = ( a ⊳ s ) ⊕ ( b ⊳ s ) for all a, b ∈ A and s ∈ S ,(ii) a ⊳ st = ( a ⊳ s ) ⊳ t for all a ∈ A and s, t ∈ S ,(iii) a ⊳ e = a ⊕ e for all a ∈ A and e ∈ E ( S ) ,(iv) e ⊳ s = 0 s − es for all e ∈ E ( S ) and s ∈ S .A free S –module F = ( F e , φ ef ) has as basis a family of sets B = { B e : e ∈ E ( S ) } , and F e is the free abelian group on the set { ( b, s ) : b ∈ B f , s ∈ S, f > ss − , s − s = e } , with ( b, s ) φ ee ′ = ( b, se ′ ) and with S –action defined by ( b, s ) ⊳ t = ( b, st ) , see [15, section3]. Lemma 1.5.
Let ψ : T → S be a surjective idempotent separating homomorphism withkernel K . Then K = F e ∈ E ( T ) K abe is an S –module, with S –action defined by k ⊳ s = t − kt for any t ∈ T with s = tψ . Crossed modules of groupoids.
We now present the rudiments of the theory ofcrossed modules of groupoids. For further information we refer to [5], and for the useof crossed modules in the theory of group presentations to [2].
Definition 1.3.
Let G be a groupoid with vertex set V = V ( G ) . Then a crossed G -module C ∂ −→ G ⇒ V EWRITING FOR PRESENTATIONS OF INVERSE MONOIDS 5 consists of:(1) a disjoint union of groups C = F e ∈ V C e , indexed by V ,(2) a homomorphism ∂ of groupoids,(3) an action of G on C , denoted ( c, g ) c g , such that an edge g ∈ G with g d = e and g r = f , acts on c ∈ C e with c g ∈ C f .The action of G on C satisfies ( c g ) ∂ = g − ( c∂ ) g whenever c g is defined,(1.1) c a∂ = a − ca where, for some e ∈ V, a, c ∈ C e . (1.2) Definition 1.4.
Consider a crossed G –module C ∂ −→ G ⇒ V along with a set R and a function ω : R → G such that ω d = ω r . Then C is said to be the free crossed G -module on ω if for any crossed G –module C ′ ∂ ′ −→ G ⇒ V and function σ : R → C ′ such that ω = σ∂ ′ there exists a unique morphism of crossed G –modules φ : C → C ′ such that ∂ = φ∂ ′ .We sketch the construction of free crossed modules: see [5, section 7.3]. Proposition 1.6. [5, Proposition 7.3.7]
Given a groupoid G , a set R and a function ω : R → G such that ω d = ω r , then a free crossed G -module on ω exists and is unique up toisomorphism.Proof. For each e ∈ V ( G ) we define R e = { s ∈ R : ( sω ) r = e = ( sω ) d } and ( R ≬ G ) e = { ( s, g ) ∈ R × G : r ∈ R gg − , g − g = e } . We define F e to be the free group on ( R ≬ G ) e , and F = F e ∈ V F e . Then we havea map δ : F → G , defined on generators by ( s, g ) g − ( sω ) g , and an action of G on F , defined on generators by ( s, g ) h = ( s, gh ) whenever g − g = hh − . We let P e denote the subgroup of F e generated by the elements of the form h u, v i = u − v − uv uδ ,for u, v ∈ F e . Then P e is normal in F e , invariant under the G –action, and contained in thekernel of δ . So δ induces ∂ : F e ∈ V F e /P e → G and this is a free crossed module on ω .Uniqueness follows from the usual universal argument. (cid:3) We note that there exists a function ν : R → C induced by mapping s ∈ R e to ( s, e ) ∈ ( R ≬ G ) e , and that ν∂ = ω .1.5.1. Modules and crossed modules.
Definition 1.5.
Consider a crossed module C ∂ −→ G ⇒ V in which ∂ is trivial: that is, ∂ maps each a ∈ C e to e ∈ V . We write ∂ = ε . By CM2 each C e is then abelian, and C isa G –module . The concept of a free G –module then follows: given a set R and a function ω : R → G with ω d = ω r , a G –module A is free on ω , if for any G –module B andfunction ν : R → B such that νε = ω , there exists a unique morphism φ : A → B of G –modules.More generally, we have: REWRITING FOR PRESENTATIONS OF INVERSE MONOIDS
Proposition 1.7. (1) Let C ∂ −→ G ⇒ V be a crossed G –module, and let Q be the quotient groupoid G/ C ∂ , with π : G → Q the natural map. Then C ab = F e ∈ V C abe is a Q –module,where for c ∈ C e and q = gπ with gg − = e we have c ⊳ q = c g . (2) If C ∂ −→ G ⇒ V is a free crossed module with basis ω : R → G then C ab is a free Q –module with basis the image of the induced map R → C → C .Proof. The claimed Q –action is well-defined, since if q = gπ = hπ with g, h ∈ G , then h = ( a∂ ) g for some a ∈ C , and then by CM2, c h = c ( a∂ ) g = ( a − ca ) g = c g . Now let A be an arbitrary Q -module, and consider the disjoint union of groups Λ = ⊔ e ∈ V Λ e , where Λ e = C e ∂ × A e . We let G act on Λ by conjugation on each C e and via π on A e . Let p : Λ → G be the projection map: we claim that Λ p −→ G ⇒ V is a crossed G –module. For ( c∂, a ) ∈ Λ e and g ∈ G with g d = e we have: (( c∂, a ) g ) p = ( g − ( c∂ ) g, a gπ ) p = g − ( c∂ ) g = g − ( c∂, a ) p g , and for ( c ∂, a ) , ( c ∂, a ) ∈ Λ e , ( c ∂, a ) ( c ∂,a ) p = ( c ∂, a ) c ∂ = (( c ∂ ) − ( c ∂ )( c ∂ ) , a ( c ∂ ) π )= (cid:0) ( c ∂ ) − ( c ∂ )( c ∂ ) , a (cid:1) since c ∂π = e , and = ( c ∂, a ) − ( c ∂, a )( c ∂, a ) , since A e is abelian. So Λ p −→ G ⇒ V is a crossed G –module.Now given ν ′ : R → A , we define ν ′′ = ( ν∂, ν ′ ) : R → Λ . We note that ν ′′ p = ν∂ , and so by freeness of C , there is an induced morphism λ : C → Λ of crossed G –modules, with νλ = ν ′′ . Composing λ with the second projection p : Λ →A gives a morphism C → A that factors through C ab → A , and is easily seen to be a mapof Q –modules.The maps used in the proof are illustrated below. EWRITING FOR PRESENTATIONS OF INVERSE MONOIDS 7 C GR Λ G A ∂λν ν ′′ ν ′ ω p p (cid:3)
2. S
EMIREGULAR AND PSEUDOREGULAR GROUPOIDS
We now introduce some additional structure on a groupoid, originating in work of Brownand Gilbert [3], and further developed by Gilbert in [9] and by Brown in [4].
Definition 2.1.
Let G be a groupoid, with object set V ( G ) and domain and range maps d , r : G → V ( G ) . Then G is semiregular if • V ( G ) is a monoid, with identity e ∈ V ( G ) , • there are left and right actions of V ( G ) on G , denoted x ⊲ α , α ⊳ x , which for all x, y ∈ V ( G ) and α, β ∈ G satisfy:(a) ( xy ) ⊲ α = x ⊲ ( y ⊲ α ) ; α ⊳ ( xy ) = ( α ⊳ x ) ⊳ y ; ( x ⊲ α ) ⊳ y = x ⊲ ( α ⊳ y ) ,(b) e ⊲ α = α = α ⊳ e ,(c) ( x ⊲ α ) d = x ( α d ) ; ( α ⊳ x ) d = ( α d ) x ; ( x ⊲ α ) r = x ( α r ) ; ( α ⊳ x ) r = ( α r ) x ,(d) x ⊲ ( α ◦ β ) = ( x ⊲ α ) ◦ ( x ⊲ β ) ; ( α ◦ β ) ⊳ x = ( α ⊳ x ) ◦ ( β ⊳ x ) , whenever α ◦ β is defined,(e) x ⊲ y = 1 xy = 1 x ⊳ y .From [9, section 1] we have the following facts. Proposition 2.1. (a) Let G be a semiregular groupoid. Then there are two everywhere defined binary oper-ations on G given by: α ∗ β = ( α ⊳ β d ) ◦ ( α r ⊲ β ) α ⊛ β = ( α d ⊲ β ) ◦ ( α ⊳ β r ) . Each of the binary operations ∗ and ⊛ make G into a monoid, with identity e .(b) The binary operation ∗ and the monoid structure on V ( G ) make the semiregulargroupoid G into a strict monoidal groupoid if and only if the operations ∗ and ⊛ on G coincide. Definition 2.2.
In view of part (c) of Proposition 2.1, we say that a semiregular groupoid is monoidal if the operations ∗ and ⊛ coincide. (Brown [4] calls such semiregular groupoids commutative whiskered groupoids .) REWRITING FOR PRESENTATIONS OF INVERSE MONOIDS
Pseudoregular groupoids.
In considering presentations of inverse monoids, we shallwant to consider semiregular groupoids in which the vertex set is an inverse monoid.
Definition 2.3.
A semiregular groupoid G is a pseudoregular groupoid if V ( G ) is aninverse monoid.The name pseudoregular is chosen to reflect the close structural connection between in-verse monoids and pseudogroups, which are inverse semigroups of partial homeomor-phisms of topological spaces (see [16, section 1.1]).In a pseudoregular groupoid G , the operations, ∗ and ⊛ given in proposition 2.1(a) eachmake G into a monoid, but not necessarily an inverse monoid, as we show in the nextexample. Example 2.2.
Let ∂ : T → G be a crossed module of groups. Add a zero to G to formthe inverse semigroup G and let ∈ G act on T as the trivial endomorphism t .The product G × T is then a pseudoregular groupoid, with the following structure. Thesubset G × T is a semiregular groupoid (see [9, Proposition 1.3(ii)]) with vertex set G ,with ( g, t ) d = g and ( g, t ) r = g ( t∂ ) , and with composition ( g, t )( h, u ) = ( g, tu ) definedwhen h = h ( t∂ ) . For the additional arrows in { } × T we define (0 , t ) d = 0 = (0 , t ) r and composition (0 , t ) ◦ (0 , u ) = (0 , tu ) and so the local group at is a copy of T . Theleft and right actions of ∈ G are given by: ⊲ ( g, t ) = (0 , g, t ) = (0 , t ) , ( g, t ) ⊳ g, t )(0 ,
1) = (0 , , ⊲ (0 , t ) = (0 , , t ) = (0 , t ) , (0 , t ) ⊳ , t )(0 ,
1) = (0 , . Then ∂ : T → G is a crossed monoid (originally mono¨ıde crois´e ) in the sense of Lavend-homme and Roisin [17, Example 1.3C], and G × T is a pseudoregular groupoid G , withvertex set G . The ∗ –operation on G recovers the semidirect product G ⋉ T : ( g, t ) ∗ ( h, u ) = ( gh, t h u ) and the operations ∗ and ⊛ coincide, but G ⋉ T is not inverse. This follows from theresults of [20], but can also be seen directly, as follows.For any t ∈ T , the element (0 , t ) is an idempotent in ( G ⋉ T, ∗ ) : (0 , t ) ∗ (0 , t ) = (0 , t t ) = (0 , t ) = (0 , t ) . But for distinct s, t ∈ T we have (0 , s ) ∗ (0 , t ) = (0 , s t ) = (0 , t ) and (0 , t )(0 , s ) = (0 , s ) and so the idempotents in G ⋉ T do not commute. Since G ⋉ T is a subgroup of G ⋉ T and the other elements are idempotents, G ⋉ T is regular (and indeed orthodox , since E ( G ⋉ T ) is a subsemigroup).In a pseudoregular groupoid G , it is natural to consider star e ( G ) for each idempotent e ∈ V ( G ) . The operation ∗ then makes star e ( G ) into a semigroup. However, as thefollowing example shows, the identity arrow e at e is not necessarly an identity elementfor (star e ( G ) , ∗ ) . EWRITING FOR PRESENTATIONS OF INVERSE MONOIDS 9
Example 2.3.
Let E be the semilattice { , e, f, } with ef = 0 and consider the simplicialgroupoid E × E with vertex set E , and d and r given by the projection maps. Let U bethe subgroupoid of E × E defined by U = { ( x, y ) ∈ E × E : x = 1 = y } ∪ { (1 , } . Right and left actions of E on E × E are defined by multiplication: x ⊲ ( y, z ) = ( xy, xz ) and ( x, y ) ⊳ z = ( xz, yz ) , making U pseudoregular. The ∗ –operation is given by ( u, v ) ∗ ( x, y ) = (( u, v ) ⊳ x )( v ⊲ ( x, y )) = ( ux, vx )( vx, vy ) = ( ux, vy ) . The star at is star = { (0 , e ) , (0 , f ) , (0 , } , but the identity arrow = (0 , is not anidentity element in (star , ∗ ) .We can, however, remedy the problem illustrated in Example 2.3 by passing to a subsemi-group that does admit e as an identity. For an idempotent e ∈ V ( G ) we define star ⊲⊳e ( G ) = { e ⊲ α ⊳ e : α ∈ star e ( G ) } . It is clear that the operation ∗ now makes star ⊲⊳e ( G ) into a monoid with identity e . Therange map r : G → V ( G ) restricts to a semigroup morphism r e : star ⊲⊳e ( G ) → V ( G ) whose image is a monoid K e with identity e . For α ∈ star e ( G ) we set α ⊲⊳ = e ⊲ α ⊳ e and define π ⊲⊳e ( G ) = { α ⊲⊳ ∈ star ⊲⊳e ( G ) : ( α ⊲⊳ ) r = e } . Proposition 2.4.
In a pseudoregular groupoid G , the binary operation ∗ and the groupoidcomposition ◦ coincide on π ⊲⊳e ( G ) and under each operation π ⊲⊳e ( G ) is a group. Fur-thermore if G is monoidal, then π ⊲⊳e ( G ) is abelian, and the family of abelian groups π ⊲⊳ ( G ) = { π ⊲⊳e ( G ) , e ∈ E ( V ( G )) } , is a V ( G ) –module.Proof. For α ⊲⊳ , β ⊲⊳ ∈ π ⊲⊳e ( G ) we have α ⊲⊳ ∗ β ⊲⊳ = ( α ⊲⊳ ⊳ ( β ⊲⊳ ) d ) ◦ (( α ⊲⊳ ) r ⊲ β ⊲⊳ ) = ( α ⊲⊳ ⊳ e ) ◦ ( e ⊲ β ⊲⊳ ) = α ⊲⊳ ◦ β ⊲⊳ . Since e ⊲ α − ⊳ e = ( e ⊲ α ⊳ e ) − it is clear that π ⊲⊳e ( G ) is a subgroup of the local group π ( G, e ) at e in the groupoid G .If G is monoidal, then ∗ and ⊛ coincide,and α ⊲⊳ ◦ β ⊲⊳ = α ⊲⊳ ∗ β ⊲⊳ = α ⊲⊳ ⊛ β ⊲⊳ = ( e ⊲ β ⊲⊳ ) ◦ ( α ⊲⊳ ⊳ e ) = β ⊲⊳ ◦ α ⊲⊳ . So π ⊲⊳e is abelian. Now for e > f we define ϕ ef : π ⊲⊳e ( G ) → π ⊲⊳f ( G ) by α ⊲⊳ f ⊲ α ⊲⊳ ⊳ f ∈ π ⊲⊳f . Then for α ⊲⊳ , β ⊲⊳ ∈ π ⊲⊳e : ( α ⊲⊳ ∗ β ⊲⊳ ) ϕ ef = f ⊲ ( α ⊲⊳ ∗ β ⊲⊳ ) ⊳ f = f ⊲ ( α ⊲⊳ ◦ β ⊲⊳ ) ⊳ f = ( f ⊲ α ⊲⊳ ⊳ f ) ◦ ( f ⊲ β ⊲⊳ ⊳ f )= ( f ⊲ α ⊲⊳ ⊳ f ) ∗ ( f ⊲ β ⊲⊳ ⊳ f )= α ⊲⊳ ϕ ef ∗ β ⊲⊳ ϕ ef and so each ϕ ef is a homomorphism. Furthermore, if e > f > g then α ⊲⊳ ϕ ef ϕ fg = g ⊲ ( f ⊲ α ⊲⊳ ⊳ f ) ⊳ g = gf ⊲ α ⊲⊳ ⊳ f g = g ⊲ α ⊲⊳ ⊳ g = α ⊲⊳ ϕ eg and clearly ϕ ee is the identity on π ⊲⊳e . Therefore ( π ⊲⊳e , ϕ ef ) is a presheaf of abelian groupsand a V ( G ) –action is now given by α ⊳ s = s − ⊲ α ⊳ s ∈ π ⊲⊳s − es . It is easy to checkthat the conditions in Definition 1.2 for a Lausch V ( G ) –module are satisfied. (cid:3)
3. T
HE RELATION MODULE OF AN INVERSE MONOID PRESENTATION
Let G be a group generated by a set X , with corresponding presentation map θ : F ( X ) → G . Let N be the kernel of θ : then conjugation in F ( X ) induces a G –action on the abelian-isation N ab of N , and N ab is the relation module . As shown in [2, Corollary 5.1], therelation module is isomorphic to the first homology group of the Cayley graph Cay(
G, X ) .In [10] the first author introduced relation modules for inverse monoid presentations byadapting work of Crowell [7] on group presentations. It was remarked in [10] thatDefining the relation module in this way permits the introduction of theconcept in other algebraic settings where the operation of abelianisationhas no obvious counterpart.However, it turns out (as we shall see below) that we can indeed define the relation moduleof an inverse monoid presentation as the abelianisation of a certain Clifford semigroup, ina precise analogy of the construction for groups. We first describe a factorization result forinverse semigroups homomorphisms. Our discussion is based on [18, page 265], to whichwe refer for further details. The result originates in [23, Theorem 4.2]. Proposition 3.1.
Let ρ be a congruence on the inverse semigroup S . Then there exists asmallest congruence ρ min on S whose trace is the same as the trace of ρ , defined by (3.1) a ρ min b ⇐⇒ there exists e ∈ E ( S ) with ae = be and a − a ρ e ρ b − b . Furthermore,(a) For a, b ∈ S we have (3.2) a ρ min b ⇐⇒ there exists c ∈ S with a > c b and a ρ c ρ b. (b) The canonical map ψ : S/ρ min → S/ρ is idempotent separating,(c) If S is E –unitary then the canonical map τ : S → S/ρ min is idempotent pure.Proof. (a) We first show that the conditions (3.1) and (3.2) are equivalent. First assumethat (3.1) holds and set c = ae = be . Then a > c b and, since a − a ρ e ρ b − b we have a = aa − a ρ ae = be ρ bb − b = b . Now if (3.2) holds, take e = c − c . Since a > c b we have ae = c = be , and since a ρ c we have a − a ρ c − c = e . Similarly b − b ρ c .(b) Suppose that a, b ∈ S with a ρ min a and b ρ min b . By Lallement’s Lemma [13,Lemma 2.4.3], there exist e, f ∈ S with a ρ min e and b ρ min f . If now e ρ f then e ρ min f , and so a ρ min b . Hence ψ is idempotent separating.(c) Suppose that, for s ∈ S and x ∈ E ( S ) , we have s ρ min x . Then there exists e ∈ E ( S ) with se = xe and xe ∈ E ( S ) , and if S is E –unitary, we have s ∈ E ( S ) and so τ isidempotent pure. (cid:3) EWRITING FOR PRESENTATIONS OF INVERSE MONOIDS 11
We now consider an inverse monoid presentation P = [ X : R ] of an inverse monoid M .We set A = X ⊔ X − , and so M is then a quotient of the free monoid A ∗ , with canonicalmap ϕ : A ∗ → M , and also a quotient of the free inverse monoid FIM( X ) , with associatedpresentation map θ : FIM( X ) → M . The Wagner congruence on A ∗ induces the naturalmap ρ : A ∗ → FIM( X ) , and ϕ = ρθ , and we may factorize θ as in Proposition 3.1. Weset T ( M, X ) = FIM( X ) /θ min and so have the commutative diagram(3.3) A ∗ ρ / / ϕ % % FIM( X ) τ / / θ T ( M, X ) ψ / / M Since
FIM( X ) is E –unitary, the map τ is idempotent pure and we obtain from [18] thefollowing structural information on T ( M, X ) . Lemma 3.2. [18, Lemma 1.6]
Let P = [ X : R ] be a presentation of an inverse monoid T .Then the following are equivalent:(a) the presentation map θ : FIM( X ) → T is idempotent pure.(b) P is equivalent to a presentation of the form P = [ X ; R ] where R = { e i = f i : i ∈ I } for some set I and idempotents e i , f i of FIM( X ) .(c) Each Sch¨utzenberger graph, Sch L ( T, X, e ) is a tree. Definition 3.1.
An inverse monoid T is arboreal [10] if it satisfies the conditions ofLemma 3.2. Corollary 3.3.
An arboreal inverse monoid T is E –unitary.Proof. It follows from part (b) of Lemma 3.2 that T has maximum group image F ( X ) andthat the quotient map σ : FIM( X ) → F ( X ) factorizes as τ σ T . Since σ is idempotent pureand τ is surjective, the map σ T is idempotent pure. (cid:3) The factorization of θ shown in (3.3) gives us an idempotent separating homomorphism ψ : T ( M, X ) → M . By Proposition 1.4, K = ker ψ = { w ∈ T ( M, X ) : wψ ∈ E ( M ) } . is a Clifford semigroup, and so is a union of groups K e , indexed by the idempotents of M .Hence K has the natural abelianisation K = [ e ∈ E ( M ) K abe that is an M –module by Lemma 1.5. Definition 3.2.
The relation module of the presentation P is the M –module K .We now draw the connection between relation modules and Sch¨utzenberger graphs. Forthe left Sch¨utzenberger graph Sch L ( M, X ) , the cellular chain group C (Sch L ( M, X, e )) is the free abelian group on the L –class L e in M , and C (Sch L ( M, X, e )) is the freeabelian group on the set { ( x, s ) : x ∈ X, s ∈ M, ( x − x ) θ > ss − , s − s = e } .2 REWRITING FOR PRESENTATIONS OF INVERSE MONOIDS
The relation module of the presentation P is the M –module K .We now draw the connection between relation modules and Sch¨utzenberger graphs. Forthe left Sch¨utzenberger graph Sch L ( M, X ) , the cellular chain group C (Sch L ( M, X, e )) is the free abelian group on the L –class L e in M , and C (Sch L ( M, X, e )) is the freeabelian group on the set { ( x, s ) : x ∈ X, s ∈ M, ( x − x ) θ > ss − , s − s = e } .2 REWRITING FOR PRESENTATIONS OF INVERSE MONOIDS The boundary map ∂ : C (Sch L ( M, X, e )) → C (Sch L ( M, X, e )) maps ( x, s ) ( xθ ) s − s . Now C (Sch L ( M, X )) = M e ∈ E ( M ) C (Sch L ( M, X, e )) and C (Sch L ( M, X )) = M e ∈ E ( M ) C (Sch L ( M, X, e )) , and by defining s ⊳ t = st and ( a, s ) ⊳ t = ( a, st ) we get an M –module structure on each of C (Sch L ( M, X )) and C (Sch L ( M, X )) . Theboundary map ∂ : C (Sch L ( M, X )) → C (Sch L ( M, X )) is then a map of M –modules,and its kernel H (Sch L ( M, X )) is an M –module H , with the group H (Sch L ( M, X )) e being the first homology group H (Sch L ( M, X, e )) of the connected component contain-ing e ∈ E ( M ) .For e ∈ E ( M ) we define U e = { w ∈ A ∗ : wϕ > e } . Then U e is a submonoid of A ∗ and is the reverse of the language accepted by the Sch¨utzenberger graph Sch L ( M, X, e ) when regarded as an automaton with input alphabet A and unique start/accept state e , see[27]. (The reversal arises because we assume that as an automaton, Sch L ( M, X, e ) readsinput words from the left, but the action on states is by left multiplication in M .) Let π e denote the fundamental group π (Sch L ( M, X, e ) , e ) . A closed path α in Sch L ( M, X, e ) is labelled by a unique word w ∈ A ∗ whose reverse w R is in U e : we write [ w ] for thehomotopy class of α in π e . Lemma 3.4.
There is a group isomorphism κ e : π e → K e mapping the homotopy class [ w ] ∈ π e to [( w R ρτ )˜ e ] − , where w R is the reverse of w and ˜ e is the unique preimage in E ( T ( M, X )) of e ∈ E ( M ) .Proof. If [ w ] ∈ π e then w R ∈ U e and ψ : ( w R ρτ )˜ e ( w R ρτ ψ )(˜ eψ ) = ( w R ϕ ) e = e . Hence [( w R ρτ )˜ e ] − ∈ K e . To verify that κ e is well-defined on π e , suppose that [ u ] = [ v ] .Then v can be obtained from u by the insertion and deletion of subwords aa − with a ∈ A .Considering one such step, if for some p, q ∈ A ∗ we have u = pq and v = paa − q then [( u R ρτ )˜ e ] − > [( v R ρτ )˜ e ] − in the subgroup K e of T ( M, X ) : since the relation > is trivial on K e , we deduce that [( u R ρτ )˜ e ] − = [( v R ρτ )˜ e ] − , and so κ e is well-defined.Now for u, v ∈ U e we have [ u ] · [ v ] = [ uv ] [(( uv ) R ρτ )˜ e ] − = ˜ e ( u R ρτ ) − ( v R ρτ ) − = ˜ e ( u R ρτ ) − ˜ e ( v R ρτ ) − = ([ u ] κ e )([ v ] κ e ) , since ˜ e is the identity of the group K e and ˜ e ( u R ρτ ) − ∈ K e . Hence κ e is a homomor-phism.If k ∈ K e we set w k to be any word in A ∗ with w R k ρτ = k . Then e = w R k ρτ ψ = w R k ϕ and so w R k ∈ U e and [ w k ] ∈ π e . Therefore κ e is surjective. EWRITING FOR PRESENTATIONS OF INVERSE MONOIDS 13
Now suppose that [ w ] ∈ ker κ e . Then ( w R ρτ )˜ e = ˜ e ∈ T ( M, X ) , and since T ( M, X ) is E –unitary, we deduce that w R ρτ ∈ E ( T ) . Since τ is idempotent pure, w R ρ ∈ E (FIM( X )) and w R is freely reducible to the empty word: hence the circuit at e in Sch L ( M, X, e ) la-belled by w is homotopic to the constant path at e , and [ w ] is trivial. Therefore κ e isinjective. (cid:3) Theorem 3.5.
The M –modules K = G e ∈ E ( M ) K abe and H = G e ∈ E ( M ) H (Sch L ( M, X, e )) are isomorphic.Proof. We identify H (Sch L ( M, X, e )) with π abe to exploit Lemma 3.4: for each e ∈ E ( M ) there is a group isomorphism ¯ κ e : π abe → K abe . The action of t ∈ M on H isinduced by the family of maps π abe → π abt − et , in which the image of a closed path [ w ] in π abe is mapped to the image of [ u − wu ] in π abt − et , where t − = u R ϕ (and so u labels apath from et to t − et in Sch L ( M, X, t − et ) ). We note that the isomorphism κ t − et maps [ u − wu ] ( u − wu ) R ρτ · ] t − et , where ] t − et is the unique element of E ( T ( M, X )) with ( ] t − et ) ψ = t − et .By Lemma 1.5, the M –action on K is induced by conjugation in T ( M, X ) : for k ∈ K e with image ¯ k ∈ K abe , ¯ k ⊳ t = ˜ t − k ˜ t ∈ K abt − et for any ˜ t with ˜ tψ = t . We set ˜ t = ( u − ) R ρτ . Then for [ w ] ∈ π e , ˜ t − ([ w ] κ e )˜ t = ˜ t − ( w R ρτ )˜ e ˜ t = ˜ t − ( w R ρτ )˜ t ˜ t − ˜ e ˜ t = ( u R ρτ )( w R ρτ )(( u − ) R ρτ )˜ t − ˜ e ˜ t = (( u − wu ) R ρτ ) · ] t − et using the fact that ψ is idempotent separating. Therefore the diagram π abe ¯ κ e / / (cid:15) (cid:15) K abe ⊳ t (cid:15) (cid:15) π abt − et ¯ κ t − et / / K abt − et commutes, and the family of maps { ¯ κ e : E ∈ E ( M ) } is an M –module isomorphism. (cid:3) Examples of relation modules.
Example 3.6.
Let M be the semilattice { , e, f, ef } , generated as an inverse monoid by { e, f } . The Sch¨utzenberger graph is • e • • fef • e fe f and the relation module is therefore Z ZZ ⊕ Z where all the structure maps are inclusions. Example 3.7.
The bicyclic monoid B is the inverse monoid presented by [ x : xx − = 1] .The Sch¨utzenberger graph Sch L ( B, x, x − q x q ) is the semi-infinite path x q x − x q x − x q . . . x − k x q . . . x x x x x The relation module K is therefore trivial. This is no surprise: B is an arboreal inversemonoid, and this Example illustrates Lemma 3.2. Example 3.8.
Given an inverse monoid M with presentation [ Y : R ] , we add a zero to M to obtain M . For M we take the generating set X = Y ∪ { z } (with z Y ), and we havea presentation Q of M given by Q = [ Y, z : R, z = z, yz = z = zy ( y ∈ Y )] . In the Sch¨utzenberger graph there is a loop at for each element of X . If [ Y : R ] hasrelation module K then the relation module of Q can be thought of schematically as K Z | X | ζ where the map ζ carries a circuit in Sch L ( M, Y ) labelled by a word w ∈ ( Y ⊔ Y − ) ∗ tothe element of Z | Y | ⊂ Z | X | determined by w . Example 3.9.
The symmetric inverse monoid I on the set { , } is generated by thetransposition τ and the identity map ε on { } : then ετ ε is the empty map , and I = EWRITING FOR PRESENTATIONS OF INVERSE MONOIDS 15 { , τ, ε, τ ε, ετ, τ ετ, } . The Sch¨utzenberger graph
Sch L ( I , { τ, ε } ) is τε τ ε τ ετ ετ ττε ττ τ εττ ε The relation module is therefore Z τ Z ε ⊕ Z τ Z τ ⊕ Z ε Z ε ⊕ Z τ
4. T HE S QUIER COMPLEX OF AN INVERSE MONOID PRESENTATION
In this section we show that we can obtain a presentation of the relation module K , derivedfrom an inverse monoid presentation P = [ X : R ] with presentation map θ : FIM( X ) → M , from a free crossed module that is in turn derived from a Squier complex Sq( P ) asso-ciated to P . Definition 4.1.
Let P = [ X : R ] be an inverse monoid presentation of M , with presenta-tion map θ : FIM( X ) → M factorised as in (3.3), as F IM ( X ) τ −→ T ( M, X ) ψ −→ M with τ idempotent pure and ψ idempotent separating. We write T for T ( M, X ) : then the Squier complex
Sq( P ) of P is the –complex constructed as follows. • The vertex set is T . • The edge set consists of all -tuples ( p, l, r, q ) with p, q ∈ T and ( l, r ) ∈ R .Such an edge will start at p ( lρτ ) q and end at p ( rρτ ) q , so each edge correspondsto the application of a relation from P in T . An edge path in Sq( P ) thereforecorresponds to a succession of such applications. • The 2-cells correspond to applications of non-overlapping relations, and so a 2-cellis attached along every edge path of the form: • ( p ( lρτ ) qp ′ , l ′ ,r ′ , q ′ ) (cid:15) (cid:15) ( p, l,r, qp ′ ( l ′ ρτ ) q ′ ) / / • ( p ( rρτ ) qp ′ , l ′ ,r ′ , q ′ ) (cid:15) (cid:15) • ( p, l,r, qp ′ ( r ′ ρτ ) q ′ ) / / • This attachment of –cells makes the two edge paths between p ( lρτ ) qp ′ ( l ′ ρτ ) q ′ and p ( rρτ ) qp ′ ( r ′ ρτ ) q ′ homotopic in Sq( P ) . Proposition 4.1.
The fundamental groupoid π ( Sq ( P ) , T ) is pseudoregular and monoidal.Proof. The actions of T on single edges in Sq( P ) given by t ⊲ ( p, l, r, q ) = ( tp, l, r, q ) and ( p, l, r, q ) ⊳ t = ( p, l, r, qt ) induce a pseudoregular structure on the fundamental groupoid. The –cells of Sq( P ) en-sure that, if α and β are the homotopy classes of edge-paths of length in Sq( P ) , then α ∗ β = α ⊛ β , and a straightforward induction extends this to arbitrary edge-paths. (cid:3) It will be convenient in what follows to describe operations in the fundamental groupoid π (Sq( P ) , T ) as being performed on edge-paths in Sq( P ) rather than on fixed-end-pointhomotopy classes.Let e ∈ E ( T ) . Then star ⊲⊳e ( π (Sq( P ) , T )) has vertex set K e = { a ∈ T : aψ = e } , whichwe recognise as one of the groups that make up the kernel of ψ . Lemma 4.2.
Let e ∈ E ( T ) . Then (star ⊲⊳e ( π (Sq( P ) , T )) , ∗ ) is a group.Proof. The set star ⊲⊳e ( π (Sq( P ) , T )) is a monoid under the operation ∗ , and for α ∈ star ⊲⊳e ( π (Sq( P ) , T )) we define α ∗ = ( α r ) − ⊲ α ◦ ⊳ ( α d ) − , where a superscript − denotes the inverse in the inverse monoid T and a superscript ◦ denotes the inverse in the groupoid Sq( P ) . Now α ∗ α ∗ = α ∗ (cid:0) ( α r ) − ⊲ α ◦ ⊳ ( α d ) − (cid:1) = (cid:0) α ⊳ ( α r ) − ( α ◦ d )( α d ) − (cid:1) ◦ (cid:0) ( α r )( α r ) − ⊲ α ◦ ⊳ ( α d ) − (cid:1) = (cid:0) α ⊳ ( α r ) − ( α r )( α d ) − (cid:1) ◦ (cid:0) ( α r )( α r ) − ⊲ α ◦ ⊳ ( α d ) − (cid:1) . Since α ∈ star ⊲⊳e ( π (Sq( P ) , T )) we have α d = e , and since α r ∈ K e and K e is a subgroupof T with identity e , then ( α r ) − ( α r ) = e = ( α r )( α r ) − . So α ∗ α ∗ = ( α ⊳ e ) ◦ ( e ⊲ α ◦ ⊳ e ) = α ◦ α ◦ = 1 e . Similarly α ∗ ∗ α = 1 e , and star ⊲⊳e ( π (Sq( P ) , T )) is a group. (cid:3) Since we shall be working exclusively in the groupoid π (Sq( P ) , T ) hereon, we shall ab-breviate star ⊲⊳e ( π (Sq( P ) , T )) to star ⊲⊳e . Lemma 4.3.
Suppose that ( ep, l, r, qe ) ∈ star ⊲⊳e and set h = lρτ and k = rρτ . Then ephqe = e and ( ep )( hqe )( ep ) = eep = ep and ( hqe )( ep )( hqe ) = hqee = hqe . Therefore ep = ( hqe ) − in T and ( ep, l, r, qe ) = ( eq − h − , l, r, qe ) : moreover e = eq − h − hqe and so e q − h − hq . Lemma 4.4.
A path α ∈ star ⊲⊳e can be written as a product of single edges in thegroup (star ⊲⊳e , ∗ ) . Hence star ⊲⊳e is generated by the subset Σ ⊲⊳e of homotopy classes ofsingle edges in star ⊲⊳e , and these classes are represented by edges of the form λ el,r,q =( eq − ( l − ) ρτ, l, r, qe ) with e q (( l − l ) ρτ ) q . EWRITING FOR PRESENTATIONS OF INVERSE MONOIDS 17
Proof.
The vertex set of the connected component of
Sq( P ) that contains e is the group K e (with identity e ), and for a path α in this component we define αλ = ( α d ) − ⊲ α ⊳ e . If α = ( p, l, r, q ) is a single edge and h = lρτ , then αλ = ( q − h − p − p, l, r, qe ) = ( eq − h − p − p, l, r, qe ) since phq ∈ K e . Then by Lemma 4.3, we have eq − h − p − p = ( hqe ) − and so αλ = ( eq − h − , l, r, qe ) = λ el,r,q , and ( αλ ) d = eq − h − hqe = e .Now if α = α ◦ α then αλ = ( α d ) − ⊲ ( α ◦ α ) ⊳ e = (( α d ) − ⊲ α ⊳ e ) ◦ (( α d ) − ⊲ α ⊳ e )= α λ ◦ (( α d ) − ⊲ α ⊳ e ) , and α λ ∗ α λ = ( α λ ⊳ e ) ◦ (( α d ) − ( α r ) e ( α d ) − ⊲ α ⊳ e ) . But α r = α d ∈ K e and so ( α d ) − ( α r ) e ( α d ) − = ( α d ) − and therefore αλ = α λ ∗ α λ . The Lemma then follows easily by induction on the length of a path. (cid:3) Now suppose that a path α with α d = e is a composition α = α ◦ α and that β is thepath β = α ◦ γ ◦ γ ◦ ◦ α for some path γ . Then βλ = α λ ∗ γλ ∗ γ ◦ λ ∗ α λ . Now if x = γ d and y = γ r then γλ ∗ γ ◦ λ = ( x − ⊲ γ ⊳ e ) ∗ ( y − ⊲ γ ◦ ⊳ e )= ( x − ⊲ γ ⊳ e ) ◦ ( x − y ⊲ y − ⊲ γ ◦ ⊳ e )= ( x − ⊲ γ ⊳ e ) ◦ ( x − ⊲ γ ◦ ⊳ e )= x − ⊲ ( γ ◦ γ ◦ ) ⊳ e = x − ⊲ x ⊳ e = 1 e . Hence if α and β are paths differing by a cancelling pair of edges in Sq( P ) then αλ = βλ in the group (star ⊲⊳e , ∗ ) .Now consider a –cell in the component of Sq( P ) containing e :(4.1) • ( p ( lρτ ) qt,s,d,u ) (cid:15) (cid:15) ( p,l,r,qt ( sρτ ) u ) / / • ( p ( rρτ ) qt,s,d,u ) (cid:15) (cid:15) • ( p,l,r,qt ( dρτ ) u ) / / • with α = ( p, l, r, qt ( sρτ ) u ) , β = ( p ( rρτ ) qt, s, d, u ) , γ = ( p ( lρτ ) qt, s, d, u ) ,δ = ( p, l, r, qt ( dρτ ) u ) . Then by Lemma 4.3, αλ = λ el,r,qt ( sρτ ) u , βλ = λ es,d,u = γλ and δλ = λ el,r,qt ( dρτ ) u . Hencepath homotopy induced by the above –cell in Sq( P ) is equivalent to the relation λ el,r,v ( sρτ ) u ∗ λ es,d,u = λ es,d,u ∗ λ el,r,v ( dρτ ) u (where v = qt above). These considerations show that: Proposition 4.5.
Given e ∈ E ( T ) , q ∈ T and ( l, r ) ∈ R with e q − (( l − l ) ρτ ) q , weset λ el,r,q = ( eq − ( l − ρτ ) , l, r, qe ) . Then the following are a set of defining relations forthe group (star ⊲⊳e , ∗ ) on the generating set Σ ⊲⊳e : λ el,r,v ( sρτ ) u ∗ λ es,d,u = λ es,d,u ∗ λ el,r,v ( dρτ ) u . A crossed module from an inverse monoid presentation.
As in Example 1.3, weregard T as a groupoid ~ T (although we shall drop the arrow superscript hereon) with vertexset E = E ( T ) , and we define S ⊲⊳ = G e ∈ E star ⊲⊳e ( π (Sq( P ) , T )) . Proposition 4.6. S ⊲⊳ r −→ T ⇒ E is a crossed module of groupoids.Proof. By Lemma 4.2, each star ⊲⊳e is a group, and so S ⊲⊳ is a disjoint union of groupsindexed by E , and is a groupoid with vertex set E . Then r is a groupoid homomorphism,and is the identity on E .An action of T on S ⊲⊳ is defined using the actions in the pseudoregular groupoid π (Sq( P ) , T ) as follows: for w ∈ T and α ∈ star ⊲⊳ww − we define α w = w − ⊲ α ⊳ w ∈ star ⊲⊳w − w . Then CM1 holds, since ( α w ) r = ( w − ⊲ α ⊳ w ) r = w − ( α r ) w . For CM2, since thebinary operations ∗ and ⊛ on π ( Sq ( P ) , T ) coincide by Proposition 4.1, then α ∗ β = α ⊛ β = β ◦ ( α ⊳ β r ) and β ∗ α β r = β ∗ (( β r ) − ⊲ α ⊳ β r ) = β ◦ ( α ⊳ β r ) . So α ∗ β = β ∗ α β r . Therefore S ⊲⊳ r −→ T ⇒ E is a crossed module of groupoids. (cid:3) We shall now show that the crossed module S ⊲⊳ r −→ T ⇒ E is free, and give an explicitbasis. To do this, we give a construction of a free crossed T ( M, X ) –module directly froman inverse monoid presentation P = [ X : R ] of M and show that it is isomorphic to theone in Proposition 4.6.Suppose that ( l, r ) ∈ R , and set h = lρτ and k = rρτ . Then hψ = kψ and so ( h − k ) ψ ∈ E ( M ) . Since ψ : T → M is idempotent separating, ( h − k ) ψ = xψ for a unique x ∈ E ( T ) with ( h − k ) ψ = ( h − h ) ψ = ( k − k ) ψ and ( hk − ) ψ = ( hh − ) ψ = ( kk − ) ψ . EWRITING FOR PRESENTATIONS OF INVERSE MONOIDS 19
Hence h − h = x = k − k and hh − = kk − . So for x ∈ E ( T ) we define R x = { ( l, r, x ) ∈ R × E : ( l − r ) ρτ ψ > xψ } , and consider the set R = F x ∈ E R x , along with the function ω : R → T which maps ( l, r, x ) x (( l − r ) ρτ ) x . Then ( l, r, x ) ω d = xh − kxk − hx and ( l, r, x ) ω r = xk − hxh − kx , with (( l, r, x ) ω d ) ψ = ( xh − kxk − hx ) ψ = xψ = ( xk − hxh − kx ) ψ = (( l, r, x ) ω r ) ψ . Since ψ is idempotent separating, we conclude that ( l, r, x ) ω d = ( l, r, x ) ω r . The freecrossed T -module C ∂ −→ T ⇒ E ( M ) with basis ω is then constructed as in Proposition1.6. Theorem 4.7.
The crossed T –module S ⊲⊳ r −→ T ⇒ E ( M ) is isomorphic to the freecrossed T –module C ∂ −→ T ⇒ E ( M ) .Proof. For ( l, r, x ) ∈ R , we retain the notation h = lρτ and k = rρτ . We define ν : R → S ⊲⊳ by ( l, r, x ) ( xh − , l, r, x ) . We then have ( xh − , l, r, x ) d = xh − hx = x , and so ( l, r, x ) ν ∈ star ⊲⊳x . Moreover, ( l, r, x ) ν r = ( xh − , l, r, x ) r = xh − kx = ( l, r, x ) ω . Therefore ν r = ω and by freeness of C ∂ −→ T ⇒ E ( M ) there exists a crossed modulemorphism η : C → S ⊲⊳ mapping ( l, r, u ) ( u − h − , l, r, u ) ∈ star ⊲⊳ ( u − u ) , where ( h − k ) ψ > ( uu − ) ψ and h − h = k − k > uu − . We claim that η is an isomor-phism, and we verify this by constructing its inverse.We define µ : Σ ⊲⊳e → C e by µ : λ el,r,q ( l, r, qe ) , where λ el,r,q is defined in Proposition4.5. We consider the effect of this map on a defining relation λ el,r,v ( sρτ ) u ∗ λ es,d,u = λ es,d,u ∗ λ el,r,v ( dρτ ) u . as given in Proposition 4.5. We set g = sρτ and t = dρτ . Then λ el,r,vgu µ ( l, r, vgue ) , λ es,d,u µ ( s, d, ue ) , and λ el,r,vtu µ ( l, r, vtue ) . In the group C e we have ( s, d, ue ) − ( l, r, vgue )( s, d, ue ) = ( l, r, vgueu − g − tue ) . Now gψ = tψ and so [( gue )( gue ) − ] ψ = [( tue )( tue ) − ] ψ . Since ψ is idempotent separating, ( gue )( gue ) − = ( tue )( tue ) − and therefore vgueu − g − tue = v ( gue )( gue ) − tue = v ( tue )( tue ) − ( tue ) = vtue . So in C e we have ( s, d, ue ) − ( l, r, vgue )( s, d, ue ) = ( l, r, vtue ) and µ induces a homomorphism star ⊲⊳e → C e that is the inverse of η . (cid:3) As a module for the groupoid ~M , we see by Proposition 1.7 that ( S ⊲⊳ ) ab is free, with basisfunction ( l, r, u ) ( u − l − , l, r, u ) . However, we can say more. Proposition 4.8. ( S ⊲⊳ ) ab is the free M –module on the E ( M ) –set Z in which Z e = { ( l, r ) ∈ R : ( l − r ) ρτ ψ = eψ } .Proof. The groupoid action of ~M extends to one of M . If α ∈ star ⊲⊳e with image α ∈ (star ⊲⊳e ) ab , and m ∈ M with m = wψ , then we define α ⊳ m = w − ⊲ α ⊳ w . As a component of the free ~M –module ( S ⊲⊳ ) ab , the group (star ⊲⊳e ) ab is the free abeliangroup with basis Y e = { ( l, r, m ) : ( l − r ) ρτ ψ > mm − , m − m = e } . which is the correct basis for ( S ⊲⊳ ) ab as the free M –module on the E ( M ) –set Z . (cid:3) A presentation for the relation module.
From an inverse monoid presentation P =[ X : R ] of an inverse monoid M we have now constructed a free crossed module S ⊲⊳ r −→T ⇒ E and for each e ∈ E we have a crossed module of groups star ⊲⊳e r −→ K e . Since K e is the vertex set of the component of Sq( P ) containing e , the map r : star ⊲⊳e → K e issurjective. By Propositions 2.4 and 4.1, π ⊲⊳e = { α ∈ star ⊲⊳e : α r = e } is abelian and we have a short exact sequence of groups(4.2) → π ⊲⊳e → star ⊲⊳e → K e → . Lemma 4.9.
Each group K e is free, the sequence (4.2) splits and, star ⊲⊳e and π ⊲⊳e × K e areisomorphic groups.Proof. The group K e is a subgroup of T and the maximum group image map σ : T → F ( X ) is idempotent pure. Its restriction σ : K e → F ( X ) therefore has trivial kernel andso K e is isomorphic to a subgroup of a free group and is free. By (1.2), K e acts trivially on π ⊲⊳e and so the splitting of the sequence (4.2) induces an isomorphism star ⊲⊳e ∼ = π ⊲⊳e × K e . (cid:3) Theorem 4.10.
Let P be an inverse monoid presentation of an inverse monoid M . Thereexists a short exact sequence of M –modules (4.3) → G e ∈ E ( M ) π ⊲⊳e → ( S ⊲⊳ ) ab r −→ K → in which ( S ⊲⊳ ) ab is a free M –module and r is induced by S ⊲⊳ r −→ K .Proof. The M –module structure on F e ∈ E ( M ) π ⊲⊳e is given by Proposition 2.4, that on ( S ⊲⊳ ) ab by Proposition 4.8, and that on K by Lemma 1.5. Lemma 4.9 gives us, for each e ∈ E ( M ) , a short exact sequence of abelian groups → π ⊲⊳e → star ⊲⊳e ∂ −→ K abe → and these assemble into the sequence (4.3). It remains to check that r is then a map of M –modules. EWRITING FOR PRESENTATIONS OF INVERSE MONOIDS 21
Let α ∈ star ⊲⊳e with image α ∈ ( S ⊲⊳ ) ab , and let m ∈ M with mm − = e . Then the actionof m on α is defined by lifting m to T and acting on α in the crossed module S ⊲⊳ r −→ T : α ⊳ m = t − ⊲ α ⊳ t where tψ = m . Hence ( α ⊳ m ) r = ( t − ⊲ α ⊳ t ) r = t − ( α r ) t ∈ U abm − m . But α r = α r and α r ⊳ m = t − ( α r ) t . (cid:3) Acknowledgements.
A version of these results is presented in the second author’s PhDthesis at Heriot-Watt University, Edinburgh. The generous financial support of a PhDScholarship from the Carnegie Trust for the Universities of Scotland is duly and gratefullyacknowledged. R
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