cs.biu.ac.ilmargolis/papers/invexpfinal.pdfcs.biu.ac.il

EXPANSIONS OF INVERSE SEMIGROUPS

MARK V. LAWSON, STUART W. MARGOLIS, AND BENJAMIN STEINBERG

Abstract. We construct the freest idempotent-pure expansion of aninverse semigroup, generalizing an expansion of Margolis and Meakin forthe group case. We also generalize the Birget-Rhodes prefix expansionto inverse semigroups with an application to partial actions of inversesemigroups. In the process of generalizing the latter expansion, we areled to a new class of idempotent-pure homomorphisms which we term F -morphisms. These play the same role in the theory of idempotent-purehomomorphisms that F -inverse monoids play in the theory of E-unitaryinverse semigroups.

1. Introduction

Expansions have played an important role in the semigroup theory litera-ture, yet the only widely used ‘expansions’ in inverse semigroup theory havehad as domain the category of groups: namely the prefix expansion of Birgetand Rhodes [3], and the Cayley graph expansion of the second author andMeakin [11]. In this paper, we, amongst other things, make these construc-tions true expansions by generalizing them to arbitrary inverse semigroups.Applications are given to partial actions of inverse semigroups.

The prefix expansion turns out to have a universal property in termsof what we shall call F -morphisms. These are surjective homomorphismsϕ : S → T of inverse semigroups such that tϕ−1 has a maximum for eacht ∈ T ; such morphisms are necessarily idempotent-pure; an inverse mon-oid is F -inverse if and only if its maximal group image homomorphism isan F -morphism. We show that F -morphisms are the analog of F -inversemonoids when one transfers the theory of E-unitary inverse semigroups tosurjective idempotent-pure homomorphisms. In particular, we prove that if

Date: January 20, 2002; revised November 11, 2002.1991 Mathematics Subject Classification. 20M18,20M17.Key words and phrases. Inverse semigroups, expansions, F -inverse semigroups, preho-

momorphisms, Birget-Rhodes expansion.The second author was partially funded by the Excellency Center ‘Group Theoretic

Methods in the study of Algebraic Varieties’ of the Israel Science foundation, INTASgrant 99-1224, Binational Science Foundation of Israel and the NSF. The third authorwas supported in part by NSF-NATO postdoctoral fellowship DGE-9972697, Praxis XXIscholarship BPD 16306 98, by Project SAPIENS 32817/99, by INTAS grant 99-1224, andby FCT through Centro de Matematica da Universidade do Porto. Much of this researchtook place at the Quasicrystals Workshop at the University of Essex funded by EPSRCgrant number GR/M6544.

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2 MARK V. LAWSON, STUART W. MARGOLIS, AND BENJAMIN STEINBERG

ϕ : S → T is idempotent-pure, then ϕ = jβ with j an embedding and β anF -morphism. From this, one obtains in a straightforward way Billhardt’s re-sult on factoring idempotent-pure morphisms through λ-semidirect productprojections.

2. Notation

Let A be a set and A = A ∪ A−1. We will denote by A+ (A∗) the freesemigroup (monoid) with involution on A and by FG(A) the free groupon A. If S is an A-generated inverse semigroup and w ∈ A+, then [w]Swill denote the image of w in S. If S is an inverse semigroup, E(S) willdenote the idempotents of S. In this paper, we will only deal with inversesemigroups; we leave to the reader the straightforward adjustments neededto handle the case of inverse monoids.

Inverse semigroup presentations will be written Inv〈A | R〉 where A is thegenerating set and R the set of relations.

We use Inv for the category of inverse semigroups and homomorphismswhile we use InvA for the category of A-generated inverse semigroups withhomomorphisms respecting the generators. Homomorphisms will also becalled morphisms.

Green’s relations and the natural partial order for inverse semigroups willbe used throughout; see [10, 16].

3. Expansions

An expansion of inverse semigroups is a functor F : Inv → Inv equippedwith a natural transformation η : F → 1Inv such that each componentηS : F (S) → S is surjective; expansions were introduced by Birget andRhodes [2]. In other words, an expansion assigns to each inverse semigroupS a semigroup F (S) and an onto morphism ηS : F (S) → S such that when-ever τ : S → T is a morphism, the following diagram commutes: F(S)F (τ)F (T )ηS ηT

SτT.

An expansion of inverse semigroups cut-to-generators is an assignment toeach set A of a functor FA : InvA → InvA with a natural transformationηA : FA → 1InvA

. In general, we will consider a fixed set A and drop thesubscript from the notation.

4. Schutzenberger Graphs

Recall that two elements s, t of an inverse semigroup S are said to be R-equivalent if ss−1 = tt−1. The R-class of s is denoted Rs. If S is generatedby A, then, for w ∈ A+, the Schutzenberger graph of w, S(w), has verticesR[w]S and edges of the form (s, a, t) where a ∈ A, s, t ∈ R[w]S , and sa = t.If (s, a, t) is an edge, then (t, a−1, s) will represent the same edge thought

EXPANSIONS OF INVERSE SEMIGROUPS 3

of as being traversed in the opposite direction. The idempotent [ww−1]Sis taken as the initial vertex and [w]S as the final vertex. Observe thatS(w) is dependent only on [w]S and so we can write, for s ∈ S, S(s) withoutambiguity. Also note that any twoR-equivalent elements of S have the samegraph, up to the choice of the terminal vertex, while D-equivalent elementshave isomorphic Schutzenberger graphs (here, up to the choice of initial andterminal vertices). Schutzenberger graphs were introduced by Stephen [21].

We let L(S(w)) be the collection of all words of A+ which read a pathin S(w) from the initial vertex to the terminal vertex. It is easy to checkthat v ∈ L(S(w)) if and only if [w]S ≤ [v]S in the natural partial order.Schutzenberger graphs are well known [21] to be inverse graphs meaningthat if a Schutzenberger graph has edges (s, a, t), (s, a, t′) with a ∈ A, thent = t′; hence every word in A+ labels at most one path into or out of anyvertex. In general, an inverse automaton is a connected inverse graph withdistinguished initial and terminal vertices.

We observe that if ϕ : T → S is a morphism of A-generated inverse semi-groups respecting generators then ϕ induces, for each w ∈ A+, an automatonmorphism ϕw : S(w)T → S(w)S (where the subscript indicates which semi-group is being considered). A morphism is called idempotent-pure if theinverse images of idempotents consist of idempotents. For a morphism ofA-generated inverse semigroups, this is equivalent to asking that ϕw be anembedding for each w ∈ A+. This follows from the following well-knownfacts: a morphism is idempotent-pure if and only if it is injective when re-stricted toR-classes [10]; a morphism of inverse automata is an embedding ifand only if it is injective on the vertex set [19, 21]. For the case of E-unitaryinverse semigroups, see [12].

The following key observation is due to Stephen [21].

Proposition 4.1. Let S be an A-generated inverse semigroup and v, w ∈A+. Then [v]S = [w]S if and only if L(S(v)) = L(S(w)).

In particular, if all the Schutzenberger graphs of S are finite, the wordproblem for S is solvable provided the Schutzenberger automata are con-structible; indeed, there is an algorithm to check whether two finite auto-mata accept the same language. Often this can be done; see [21].

Corollary 4.2. Let S be an A-generated inverse semigroup with finite R-classes. Then S is residually finite.

Proof. The transition semigroups of the Schutzenberger graphs are clearlyhomomorphic images of S and are finite since the R-classes are finite. Sup-pose v, w ∈ A∗ are such that [v]S 6= [w]S ; then L(S(v)) 6= L(S(w)) byProposition 4.1, so v and w cannot act the same on the vertices of bothS(v) and S(w). Thus v and w can be separated in a finite image of S. ¤


Hence, by a well-known result of Evans [6], it follows that if S has finiteSchutzenberger graphs, is recursively presented, and one can effectively enu-merate the finite inverse semigroups satisfying the defining relations of S,then one can decide the word problem for S.

The following is observed in [13].

Proposition 4.3. Suppose S = Inv〈A | R〉 where R consists of relations ofthe form tm = tm+n with n ≤ m. Then the Schutzenberger graphs of S arefinite.

This will also follow from our results (although the proof is basically thesame).

Note that if s, t ∈ S, then, since the R-relation is a left congruence,left multiplication by s on the vertex set of S(t) induces a labeled graphmorphism from S(t) to S(st) respecting the terminal vertices (though notnecessarily the initial vertices). On the other hand, left multiplication bystt−1s−1 is easily seen to induce a labeled graph morphism from S(s) to S(st)respecting the initial vertices (but not necessarily the terminal vertices); thisfollows since (st)(st)−1 = stt−1s−1 ≤ ss−1. If X is a subgraph of S(t), thensX will denote the image of X (in S(st)) under the action of s. Observethat if t ∈ s−1sS (or, equivalently, tt−1 ≤ s−1s), then left multiplication bys induces an isomorphism of labeled graphs (with inverse, left multiplicationby s−1).

Lemma 4.4. Let S be an A-generated inverse semigroup and u, v ∈ A+.Suppose that Pu is the path read in S(u) from [uu−1]S by u and Pv is thepath read in S(v) from [vv−1]S by v. Then the path read by uv in S(uv)from [uv(uv)−1]S is [uvv−1u−1]SPu ∪ [u]SPv.

Proof. From what we have so far observed, [uvv−1u−1]SPu is a path labeledby u from [uv(uv)−1]S in S(uv) whose endpoint must be [uvv−1u−1u]S =[uvv−1]S . On the other hand, uPv is a path labeled by v in S(uv) withendpoint [uv]S and hence whose starting point is [uvv−1]S . It thus followsthat the path read by uv in S(uv) is precisely [uvv−1u−1]SPu ∪ [u]SPv. ¤

We end this section with a related proposition.

Lemma 4.5. Let S be an A-generated inverse semigroup and u ∈ A+.Suppose u = vw. Then in S(u), the paths u and uw−1 use the same edges.Also, u R uw−1.

Proof. Clearly any edges used in reading u is used in reading uw−1. Whilereading u = vw from [uu−1]S to [u]S , we read a path P labeled by w from[uu−1v]S to [u]S . Hence uw−1 first traverses the path labeled by u andthen backtracks over the subpath P : no new edges are used. The secondstatement follows since

uw−1(uw−1)−1 = uw−1wu−1 = vww−1ww−1v−1 = vww−1v−1 = uu−1.

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5. Free Idempotent-Pure Expansions

In this section, we introduce an expansion cut-to-generators ( )Sl suchthat each component of the natural transformation η is idempotent-pure,and with the following universal property: if ϕ : T → S is an idempotent-pure morphism (T generated by A), then there is a unique (necessarilyidempotent-pure) morphism ψ : SSl → T such that ψϕ = ηS .

5.1. Presentations. Before giving an explicit construction, we will give apresentation for SSl and show, using Proposition 4.3, that the Schutzenbergergraphs of this semigroup are finite.

Let S = Inv〈A | R〉. Then SSl has presentation Inv〈A | R′〉 whereR′ consists of all relations u = u2 with [u]S an idempotent; if ρ is thecongruence associated to S, then the congruence associated to SSl is denotedρmin in [16]. Clearly, the congruence generated by R′ is contained in thatof R, so ηS : SSl → S exists. Since the image under a morphism of anidempotent is an idempotent, it is clear that ( )Sl is a functor and η anatural transformation. The finiteness of the Schutzenberger graphs nowfollows from Proposition 4.3. By construction, ηS is idempotent-pure anduniversal for this property. Note that if S has decidable word problem, theabove presentation is recursive.

We would like to obtain an explicit representation of this expansion (thiswas done for the case that S is a group by the second author and Meakin[11]). This will allow us to better understand Green’s relations for theresulting inverse semigroup SSl as well as to see such properties as, forinstance, the finiteness of S and A implies the finiteness of SSl. This canbe proved by Brown’s theorem [5], but our representation will make this, aswell as the finiteness of the Schutzenberger graphs in general, evident.

First we describe explicitly the congruence on A+ giving rise to SSl.

Theorem 5.1. Let S be an A-generated inverse semigroup. Define a re-lation on A+ by u ≡ v if [u]S = [v]S, and u and v use the same edges intheir respective runs from [uu−1]S = [vv−1]S in S(u) = S(v). Then ≡ is acongruence and SSl = A+/≡.

Proof. Clearly ≡ is an equivalence relation. That it is a congruence con-taining the Wagner congruence [10] follows immediately from Lemma 4.4.We now show that it has the universal property of SSl. Suppose ∼ is acongruence on A+ such that

u ∼ v =⇒ [u]S = [v]S ,

[u]S = [u]2S =⇒ u ∼ u2

Let T = A+/∼. Then the natural morphism ϕ : T → S is idempotent-pure.Suppose u, v ∈ A+ and u ≡ v. Then we have the following: ϕu : S(u)T →S(u)S is an embedding, u runs from [uu−1]T to [u]T in S(u)T , and v runsfrom [uu−1]S to [u]S in S(u)S using edges in S(u)T ϕu. Thus v ∈ L(S(u)T )


whence [u]T ≤ [v]T . A symmetric argument shows that [v]T ≤ [u]T so[u]T = [v]T . Therefore ≡⊆∼ and the result follows. ¤

Note that if S has decidable word problem, this presentation is recursive.

Corollary 5.2. Let S be an A-generated inverse semigroup and u ∈ A+ suchthat u = vw and [u]S = [uw−1]S. Then [u]SSl = [uw−1]SSl. In particular, if[u]mS = [u]m+n

S with n ≤ m, then [u]mSSl = [u]m+n

SSl .

Proof. This follows immediately from Lemma 4.5. ¤

Corollary 5.3. Let S = Inv〈A | R〉 where R consists entirely of relationsof the form u = uw−1 where u = vw, w ∈ A+. Then SSl = S, that is, S hasno non-trivial, idempotent-pure A-generated extensions. In particular, thisholds if R consists entirely of relations of the form tm = tm+n with n ≤ m.

5.2. Structure. We now describe the structure of this expansion. Let S bean A-generated inverse semigroup. Let T be the collection of inverse auto-mata over A which can be obtained as the (finite, connected) subautomatonof S(w) used in reading w from [ww−1]S to [w]S for some w ∈ A+. Let Tbe the set of all (X, s) ∈ T × S such that X has initial state ss−1 and finalstate s (so, by definition of T , X ⊆ S(s)). We define a multiplication by

(X1, s1)(X2, s2) = (s1s2s−12 s−1

1 X1 ∪ s1X2, s1s2).

From Lemma 4.4, it is clear that if X1 ∈ T corresponds to w1 ∈ A+ and X2

corresponds to w2, then s1s2s−12 s−1

1 X1 ∪ s1X2 corresponds to w1w2. Thusthe multiplication on T is well defined. It is easily checked to be associativeand the resulting semigroup is inverse: (X, s)−1 = (s−1X, s−1). In fact, theabove argument shows that the map ϕ : A+ → T defined by wϕ = (X, [w]S),where X is the collection of edges used in reading w from [ww−1]S to [w]Sin S(w), is a surjection and that the congruence associated to ϕ is ≡. ThusT = SSl. The map to S is just the projection to the second coordinate.Summing it all up, we have

Theorem 5.4. Let S be an A-generated inverse semigroup. Then SSl con-sists of all pairs (X, s) where s ∈ S and there is a word w ∈ A+ such that[w]S = s and X is the subautomaton of S(w) used in reading w from theinitial vertex. The operations are given by:

(X1, s1)(X2, s2) = (s1s2s−12 s−1

1 X1 ∪ s1X2, s1s2)

(X, s)−1 = (s−1X, s−1)

The natural transformation ηS is the projection to S; the map from A+ toSSl takes w to (X, [w]S) where X is the set of edges read by w in S(w).

For example, if S is a group, then SSl consists of all pairs (X, s) wheres ∈ S and X is a finite, connected subgraph of the Cayley graph of Scontaining 1 and s, so SSl is just the expansion of [11] applied to S.


We observe that the above inverse semigroup does not in general consistof all pairs (X, s) where X is a finite, connected subgraph of S(s) containingss−1 and s. Indeed, if S is the free inverse semigroup on x, then the pair(X, x2x−2) where X consists just of the edge (x2x−2, x, x2x−2x) (and itsreverse) is not a valid element of SSl = S as no word mapping to x2x−2 canavoid passing through the vertex x2.

In [13], it is shown that for the free inverse semigroups satisfying xn =xn+1, any finite connected inverse automaton over the same alphabet whosetransition inverse monoid satisfies xn = xn+1 can be a Schutzenberger graph.The question for the free inverse semigroups satisfying xn = x2n seemsinteresting and hard.

We now restate Theorem 5.4 in a way which makes it clearer which graphscomes up, giving a more semantic description of SSl.

Proposition 5.5. Let S be an A-generated inverse semigroup. Then SSl

consists of all pairs (X, s) where X is a finite connected subgraph of S(s)containing ss−1, s and there is a word w reading from ss−1 to s using allthe edges of X such that w cannot be read in S(t) (from tt−1 to t) for anyt > s. The operations are given by:

(X1, s1)(X2, s2) = (s1s2s−12 s−1

1 X1 ∪ s1X2, s1s2)

(X, s)−1 = (s−1X, s−1)

The natural transformation ηS is the projection to S; the map from A+ toSSl takes w to (X, [w]S) where X is the set of edges read by w in S(w).

Proof. First we show that if (X, s) is as in the theorem statement, then thereis a word w with [w]S = s and such that w uses precisely the edges of X inS(w) when reading from ss−1 to s. Indeed, let w be a word reading in Xfrom ss−1 to s using every edge of X and which cannot be read on any S(t)(from tt−1 to t) with t > s. Then [w]S ≥ s. Since w can be read on S(w),it follows [w]S = s.

Suppose now that (X, s) is a valid element of SSl as per Theorem 5.4.Choose w so that [w]S = s and w uses precisely the edges of X in S(w)when reading from ss−1 to s. Suppose there exists t > s such that w canbe read on S(t) from tt−1 to t. Then s = [w]S ≥ t, a contradiction. ¤

In our earlier example, any word readable on X is readable on S(x).In fact it is not hard to show that if S is E-unitary, then the pairs (X, s)

which arise in SSl correspond to those subgraphs X of S(s) containingss−1, s such that there is no subgraph Y ⊆ S(t) with ss−1Y = X andt > s. We do not have as satisfactory a description in general.

Observe that since ηS is idempotent-pure, the idempotents of SSl are allpairs of the form (X, e) with e ∈ E(S). Note that the initial and final statesof X will coincide.

Our coordinatization makes the following result obvious.


Corollary 5.6. Let A be finite and S be a finite A-generated inverse semi-group. Then SSl is also finite.

Let S =⋃

s∈S S(s). Then the left representation of S on itself induces anaction of S on the free semilattices F on the set of edges of S. One can thenrealize SSl as an inverse subsemigroup of the λ-semidirect product F oλ Sin an obvious way. See [1, 10] for the definitions of λ-semidirect products.

Our next goal is to characterize Green’s relations on SSl. In particular,we show that the Schutzenberger graphs of SSl are finite. For the rest ofthis section, we fix an A-generated inverse semigroup S. Since most of theresults are straightforward calculations (or one can use general results aboutλ-semidirect products or L-representations [1, 10]), we omit the proofs.

Proposition 5.7. Let (X, s), (Y, t) ∈ SSl. Then (X, s) ≤ (Y, t) if and onlyif s ≤ t and ss−1Y ⊆ X.

Now we can turn to the Green’s relations. The easiest to handle will bethe R-relation. First we remark that it is easy to see that if (X, s) ∈ SSl

and t ∈ Ver(X), then (X, t) ∈ SSl.

Proposition 5.8. Let (X, s) ∈ SSl. Then (Y, t) R (X, s) if and only ift R s and X and Y agree except for the terminal vertex.

Dually,

Proposition 5.9. Let (X, s) ∈ SSl. Then (Y, t) L (X, s) if and only if t L sand s−1X and t−1Y agree except for the terminal vertex.

The above results shows that the L and R-classes are finite. However,much more can be said.

Proposition 5.10. Let w ∈ A+. Then (ηS)w : S(w)SSl → S(w)S is anembedding with image X where (X, [w]S) = [w]SSl. In other words, S((X, s))is isomorphic to X under the map which forgets the first coordinate.

Proof. The two statements are clearly equivalent. By Proposition 5.8, theR-class of an element (X, s) consists of all pairs (Y, t) where Y is obtainedby changing the terminal vertex of X to t. Thus ηS maps the vertices ofS((X, s)) bijectively to the vertices of X. To complete the proof, we needto show that if (s, a, t) is an edge of X, then ((X, s), a, (X, t)) is an edge ofS((X, s)). But we have already seen (X, s) R (X, t) and if w ∈ A maps to(X, s), then since (s, a, t) is an edge of X ⊆ S(t), it follows that [wa]S = tand uses precisely the edges of X. Thus ((X, s), a, (X, t)) is an edge ofS((X, s)) as desired. The result follows. ¤

Corollary 5.11. For any set A and A-generated inverse semigroup S, theSchutzenberger graphs of SSl are all finite.

The H-relation has a particularly pleasant description.


Proposition 5.12. Let (X, e) be an idempotent of SSl. Then the H-classof (X, e) is isomorphic to the subgroup of He (the H-class of e) taking X toitself under left multiplication. In particular, if the maximal subgroups of Sare torsion-free, SSl is combinatorial (has only trivial subgroups).

Proof. One easily verifies that (Y, t) H (X, e) if and only if Y = X, t ∈ He

and tX = X. It now follows that the projection to S gives the desiredisomorphism. Since the R-classes of SSl are finite, so are the H-classeswhence the final statement. ¤

We remark that the D-classes of SSl are finite since the L and R-classesare finite.

5.3. Applications. We present an application to the theory of varieties.If V is a variety of inverse semigroups (a class closed under products, in-verse subsemigroups, and quotients), then Sl©m V is the variety generatedby inverse semigroups with an idempotent-pure morphism to an inversesemigroup in V. A general nonsense-type argument shows if FV(A) is therelatively free A-generated inverse semigroup in V, then FV(A)Sl is the A-generated relatively free inverse semigroup in SL©m V. Hence we can deducethe following.

Theorem 5.13. Let V be a variety of inverse semigroups, then the relativelyfree inverse semigroup in Sl©m V on any finite set has finite R-classes andSchutzenberger graphs. Furthermore, it has decidable word problem providedthe corresponding free semigroup in V does and the finite members of Sl©m Vare recursively enumerable.

Proof. It is well known [6] that a residually finite, finitely generated algebradefined by a recursive set of relations has decidable word problem if its finiteimages are recursively enumerable. Under our hypotheses, this is the casefor FV(A)Sl. ¤

Corollary 5.14. Suppose V is defined by a recursive set of equations of theform uw−1 = u, where u = vw, whose validity can be effectively tested infinite inverse semigroups; then FV(A) has decidable word problem for anyfinite set A. In particular, the variety defined by the equation tm = tm+n

with n ≤ m has these properties.

Proof. By Corollary 5.3, the variety V is fixed by the expansion ( )Sl whenceFV(A) = FV(A)Sl and hence is residually finite. The result now follows fromthe above theorem. ¤

A similar argument to the above ones shows the following.

Proposition 5.15. Let S be an inverse semigroup with a recursive pre-sentation over a finite set A consisting of relations of the form uw−1 = u,where u = vw, which can effectively be checked in finite A-generated inversesemigroups. Then S has a decidable word problem.


We mention that the expansion SGp which relates to Gp ©m V (whereGp is the variety of groups) in the same way that SSl relates to Sl©m V wasinvestigated in [13]. The reader is also referred to [17].

6. A Generalized Prefix Expansion

In [3], Birget and Rhodes introduce an expansion of semigroups called theprefix expansion. If S is a semigroup, then SPr consists of all pairs (X, s)where X is finite subset of S1 containing 1 and s and which is an ≤R-chain.The multiplication is given by (X, s)(Y, t) = (X ∪ sY, st). This semigroup isgenerated by the elements ({1, s}, s) and two words in S+ map to the sameelement of SPr if and only if they map to the same element of S and theirset of prefixes map to the same set of elements of S (whence the name).Now if G is a group, then every finite subset is an ≤R-chain and GPr isan inverse semigroup where the inverse of (X, g) is (g−1X, g−1), and theprojection ηG : GPr → G is idempotent-pure. In fact, it is easy to see that(X, g) ≤ (X ′, g′) if and only if g = g′ and X ′ ⊆ X. Thus, for each g ∈ G, theset gη−1

G has a maximum element, namely ({1, g}, g), whence GPr is whatis called an F -inverse monoid [10]. Our goal in this section is to generalizethis expansion to inverse semigroups. A word of caution: for the remainderof this paper, if S is an inverse semigroup, the notation SPr will denoteour new prefix expansion of inverse semigroups and not the Birget-Rhodesprefix expansion; in general the two only coincide on the class of groups.

6.1. Dual Prehomomorphisms and F -morphisms. First we try to mo-tivate why we wish to do this. In [22], Szendrei showed that GPr is the uni-versal F -inverse monoid with maximal group image G. That is given any F -inverse monoid I with maximal group homomorphism σ : I → G, there is aunique homomorphism ψ : GPr → I such that (max(gη−1

G ))ψ = max(gσ−1).The first author and Kellendonk [9] then showed that GPr has another

important universal property; to describe it, we must introduce some furtherconcepts. The function η∗G : G → GPr given by gη∗G = max(gη−1

G ) has theproperties:

(1) g1η∗Gg2η

∗G ≤ g1g2η

∗G;

(2) g−1η∗G = (gη∗G)−1.

Such a function between inverse semigroups is called a dual prehomomor-phism. As motivation for considering them, we mention partial actions ofan inverse semigroup on a set, a concept we now introduce. Suppose S is aninverse semigroup acting on a set Y by partial bijections and X ⊆ Y . Thenthere is a partial action of S on X by restriction. The induced map from Sto I(X) (the symmetric inverse monoid on X) is an order-preserving dualprehomomorphism. This will be touched on again in section 6.5.

It turns out that the map η∗G considered above is universal amongst alldual prehomomorphisms from G to an inverse semigroup; that is, all dual


prehomomorphisms ϕ : G → I factor through η∗G followed by a homomor-phism. This result is due to Kellendonk and Lawson [9], developing anearlier result of Exel [7]. We shall shortly see that it follows from the resultof Szendrei. Relationships between dual prehomomorphisms and F -inversemonoids can also be found in [16].

The following lemma is straightforward, but of frequent enough use to beworth isolating.

Lemma 6.1. Let θ : S → T be a dual prehomomorphism and e ∈ E(S).Then eθ ∈ E(T ).

Proof. We have the following:

eθ = eθ(eθ)−1eθ = eθeθeθ ≤ eθeθ ≤ eθ.

¤

Suppose ϕ : S → T is a surjective morphism of inverse semigroups suchthat tϕ−1 has a maximum element for each t ∈ T . Define ϕ∗ : T → Sby tϕ∗ = max tϕ−1; then ϕ∗ is a dual prehomomorphism. We call ϕ anF -morphism if ϕ∗ is order-preserving. Note that S is an F -inverse monoidif and only if the natural surjection to its maximal group image is an F -morphism. One can check that a surjective morphism ϕ is an F -morphismif and only if it has a right adjoint (viewed as a map of partially orderedsets). See [4] for more on homomorphisms of partially ordered semigroupswhich have right adjoints.

Observe that the assumption that ϕ∗ be order-preserving is not redun-dant. Indeed, let S = {x, y, xy} be a free semilattice on x and y andT = {0, 1} (with multiplication). Define ϕ : S → T by xϕ = 1, yϕ = 0.Then x = max 1ϕ−1 and y = max 0ϕ−1 but y = 0ϕ∗ � x = 1ϕ∗. Thus ϕ isnot an F -morphism.

We now aim to show that F -morphisms play the same role in the theory ofidempotent-pure homomorphisms that F -inverse monoids play in the theoryof E-unitary inverse semigroups; see [10, Chapter 7] for an exposition of this.

We mention that the F in F -morphism stands for ferme meaning closed inFrench. This is because the order-preserving dual prehomomorphism ϕϕ∗ isa closure function on S. A closure function α : S → S is an order-preservingdual prehomomorphism such that s ≤ sα and sα = sαα. Note that α is thena closure operator on S (that is, an idempotent, increasing, order-preservingendomorphism of a partially ordered set). One can then prove the followingstraightforward lemma on closure functions.

Lemma 6.2. Let α : S → S be a closure function. Then(1) s ∈ E(S) if and only if sα ∈ E(S).(2) (ab)α = (aαb)α = (abα)α = (aαbα)α.

Proof. If s ∈ E(S), then sα ∈ E(S) by Lemma 6.1. If sα ∈ E(S), thens ≤ sα, so s ∈ E(S). The second assertion is trivial. ¤


Corollary 6.3. Let ϕ : S → T be an F -morphism. Then ϕ is idempotent-pure.

Proof. Suppose sη is idempotent; then sηη∗ is an idempotent by Lemma 6.1.Since ηη∗ is a closure function, s ∈ E(S), as desired (using Lemma 6.2). ¤

If α is a closure function, then the set of closed elements Sα can be madeinto an inverse semigroup via the multiplication s · t = (st)α. Moreover,α : S → (Sα, ·) is an F -morphism with αα∗ = α. Conversely, if ϕ : S → Tis an F -morphism, then (Sϕϕ∗, ·) is isomorphic to T and the F -morphismscommute with the isomorphisms. Thus the study of F -morphisms from Sis the same as the study of closure functions on S.

To highlight the importance of F -morphisms, we prove two factorizationresults for idempotent-pure surmorphisms of inverse semigroups. The first isa new result generalizing the fact that any E-unitary inverse semigroup em-beds in an F -inverse monoid [10]; our proof, in fact, generalizes the proof ofthis older result. We then deduce as a consequence a result of Billhardt [1] onfactoring idempotent-pure homomorphisms through λ-semidirect products.

As background, we recall some classical results and definitions about par-tially ordered semigroups.

A partially ordered semigroup is, of course, a semigroup equipped with apartial ordering which is compatible with the multiplication.

Let S and T be partially ordered semigroups. McFadden [14] defines amorphism θ : S → T to be an m-homomorphism if for each a ∈ S the setaθθ−1 has a maximum element ma such that a ≤ b implies ma ≤ mb.

In what follows, every inverse semigroup is partially ordered with respectto its natural partial order. The following lemma is straightforward and weleave it to the reader.

Lemma 6.4. Let θ : S → T be a surjective homomorphism between in-verse semigroups. Then θ is an m-homomorphism if and only if it is anF -morphism.

It follows that our notion of an ‘F -morphism’ is a special case of McFad-den’s notion of an ‘m-homomorphism’.

O’Carroll [15, Theorem 3] proves the following result. Let θ : S → T be asurjective order preserving homomorphism of partially ordered semigroups S

and T . Then θ can be factored θ = jβ where Sj→ M(S)

β→ T with j an orderpreserving embedding, β a surjective order preserving m-homomorphism andM(S) a partially ordered semigroup constructed from S and T .

We would like to prove the analog of this result for inverse semigroups.However, in the case that S and T are both inverse, one finds that in general,M(S) need not be inverse. Indeed, since injections and F -morphisms areidempotent-pure, only surjective idempotent-pure maps can have a factor-ization as an inclusion followed by an F -morphism.

Thus in seeking to formulate the analog of O’Carroll’s result above, wemust restrict our attention to surjective idempotent pure homomorphisms.


We need two lemmas. Recall that a subset A of an inverse semigroup issaid to be compatible if, for all a, b ∈ A, ab−1 and a−1b are idempotents.

Lemma 6.5. Let ϕ : S → T be a morphism of inverse semigroups. Thenϕ is idempotent-pure if and only if A ⊆ T compatible, implies Aϕ−1 iscompatible.

Proof. Suppose ϕ is idempotent-pure and A ⊆ T is compatible. Let a, b ∈Aϕ−1. Then (ab−1)ϕ = aϕ(bϕ)−1 ∈ E(T ) since A is compatible. Thusab−1 ∈ E(S) since ϕ is idempotent-pure; dually a−1b ∈ E(S).

Conversely, suppose ϕ−1 preserves compatibility and e ∈ E(T ). Theneϕ−1 is a compatible set. If a ∈ eϕ−1, then so are aa−1 and a−1. Hencea = (aa−1)(a−1)−1 ∈ E(S), as desired. ¤

The following lemma is evident.

Lemma 6.6. Suppose ϕ : S → T is an F -morphism and T ′ ⊆ T is aninverse subsemigroup. Let S′ = T ′ϕ−1 and let ϕ′ : S′ → T ′ be the restriction.Then ϕ′ is an F -morphism.

Theorem 6.7. Let θ : S → T be a surjective idempotent-pure homomor-phism between inverse semigroups. Then there is an inverse semigroupM(S), an embedding j : S → M(S) and an F -morphism β : M(S) → Tsuch that θ = jβ.

Proof. We use the Schein completion C(S) of the inverse semigroup S (see[18, 10]). The inverse semigroup C(S) consists of all compatible order idealsof S. There is an embedding ιS : S → C(S) given by s 7→ [s], the principalorder ideal generated by s. Similarly, we can consider C(T ) and ι(T ). Thereader should compare with the proof of [10, Proposition 7.1.4].

Define Θ : C(S) → C(T ) by AΘ = Aθ; since θ is surjective, Aθ is acompatible order ideal. Clearly Θ is a homomorphism. Notice that θιT =ιSΘ, so Θ extends θ. We show that Θ is an F -morphism. Let B ∈ C(T ).Then Bθ−1 ∈ C(S) by Lemma 6.5. Since Bθ−1Θ = B, Θ is surjective.Moreover, we clearly have Bθ−1 = max BΘ−1.

Let M(S) = TιT Θ−1. Let us no longer distinguish SιS from S, andlikewise TιT from T . Then S ⊆ M(S) and the restriction Θ′ : M(S) → T ofΘ is an F -morphism by Lemma 6.6. Since Θ′ extends θ the result follows. ¤

Note that Theorem 6.7, when restricted to the case T is a group, showsthat each E-unitary inverse semigroup embeds in an F -inverse monoid.

As a corollary we obtain the following result of Billhardt [1].

Corollary 6.8. Let θ : S → T be a surjective idempotent-pure homomor-phism of inverse semigroups. Then there is a λ-semidirect product E oλ Twith E a semilattice, an embedding ι : S → E oλ T such that θ = ιπ whereπ is the semidirect product projection.

Proof. By Theorem 6.7, θ = jβ with j an embedding and β : M(S) →T an F -morphism. The congruence associated to β is clearly a Billhardt


congruence (see [1, 10] for the definition). Hence, by [10, Theorem 5.3.5], βcan be factored iπ where i : M(S) → KerβT oλ T is an inclusion and π isthe semidirect product projection. Setting ι = ji, we are done. ¤

This result is also proved in [8] and can be deduced from the techniquesof [20].

We thank the anonymous referee for pointing out that our Theorem 6.7can, in turn, be deduced from Bilhardt’s theorem and so the two results areformally equivalent.

We have thus far seen how we can associate an order-preserving dualprehomomorphism to any F -morphism. Now we complete the connectionby going the opposite direction.

Proposition 6.9. Let ψ : T → S be an order-preserving dual prehomomor-phism. Then there exist an inverse semigroup I, an F -morphism ϕ : I → T ,and a homomorphism ρ : I → S so that ψ = θϕρ.

Proof. Let I = {(s, t) ∈ S × T | s ≤ tψ}. We claim I is an inverse subsemi-group of S × T . Indeed, if (s1, t1), (s2, t2) ∈ I, then

t1ψt2ψ ≤ (t1t2)ψ

so s1s2 ≤ (t1t2)ψ. Closure under inverse follows easily since t−1ψ = (tψ)−1.The projection ϕ to T is an F -morphism since (tψ, t) = max(tϕ−1) bydefinition of I, and if t1 ≤ t2 ∈ T , then t1ψ ≤ t2ψ so (t1ψ, t1) ≤ (t2ψ, t2).Finally, if ρ is the projection to S, then ϕ∗ρ = ψ. ¤

One says that an F -morphism η : U → T is universal amongst F -morphisms to T if whenever there is an F -morphism ψ : S → T , thereis a unique homomorphism α : U → S with η = αψ and η∗α = ψ∗. If suchan inverse semigroup U exists, it must be unique up to isomorphism by theusual argument. One says that an order-preserving dual prehomomorphismθ : T → U is universal amongst order-preserving dual prehomomorphismsfrom T if, given any dual order-preserving prehomomorphism ψ : T → S,there is a unique homomorphism ρ : U → S such that ψ = θρ. Again, ifsuch an inverse semigroup U exists, it must be unique up to isomorphism.

Theorem 6.10. Let T be an inverse semigroup. Then if η : U → T isuniversal amongst F -morphisms to T , the associated dual prehomomorphismη∗ : T → U is universal amongst order-preserving dual prehomomorphismsfrom T .

Proof. Let ψ : T → S be a dual prehomomorphism and let I, ρ, and ϕbe as in the proof of Proposition 6.9. Then there exists a homomorphismα : U → I such that αϕ = η and η∗α = ϕ∗. So ψ = ϕ∗ρ = η∗αρ. Theuniqueness follows since if ψ = η∗β with β : U → S, then γ = β× η : U → Iis a morphism satisfying the properties demanded of α. Hence γ = α. Butβ = γρ = αρ. ¤


Specializing to the case T is a group, we have by [22] that ηT : TPr → Tis universal amongst F -morphisms to T . Hence, Theorem 6.10 shows thatTPr has the universal property proved by Kellendonk and Lawson [9]. Thatis given any dual prehomomorphism θ : T → S with S an inverse semigroup,there is a unique homomorphism ϕ : TPr → S such that η∗T ϕ = θ.

One might be tempted to prove a converse to Theorem 6.10. However,there is no need to do so. Once we show that universal F -morphisms exist, itwill follow that universal dual prehomomorphisms exist and (by uniqueness)are constructed from F -morphisms in the above fashion.

6.2. L-representations. Since F -morphisms are idempotent-pure (by Corol-lary 6.3), we can use the general theory of idempotent-pure homomorphismsto study them [10]. We begin by characterizing the L-representations of suchinverse semigroups [10]. We briefly recall McAlister-O’Carroll triples. LetS be an inverse semigroup, X a partially ordered set, and Y an order idealof X which is a semilattice in the induced order. Let I(X,≤) be the inversemonoid of order isomorphisms (acting on the left) between order ideals ofX. If P is a partially ordered set and p ∈ P , then [p] will denote the orderideal generated by p. A McAlister-O’Carroll triple (γ, q, Y ) consists of amorphism γ : S → I(X,≤) and a surjective morphism q : Y → E(S) suchthat:

(1) (Sγ)Y = X;(2) For each y ∈ Y and e ∈ E(S), y ∈ dom(eγ) if and only if yq ≤ e;(3) For each s ∈ S, there exists y ∈ Y such that yq = ss−1 and

(s−1γ)(y) ∈ Y .

Usually, we denote (sγ)(x) by sx. If (γ, q, Y ) is a McAlister-O’Carroll triple,then one obtains an inverse semigroup

L(γ, q, Y ) = {(y, s) ∈ Y × S | yq = ss−1, s−1y ∈ Y }with product given by

(x, s)(y, t) = (s(s−1x ∧ y), st).

The natural partial order turns out to be the product order. One canshow that E(L(γ, q, Y )) ∼= Y and that the projection to S is surjectiveand idempotent-pure. Conversely, see for instance [10], if ϕ : T → Sis a surjective idempotent-pure morphism, then T ∼= L(γ, q, Y ) for someMcAlister-O’Carroll triple (γ, q, Y ) and under this isomorphism, ϕ is trans-formed into the projection. One calls this an L-representation of T . SinceF -morphisms are always idempotent-pure, we can attempt to characterizethem in terms of their L-representations. In the case that S is a group, andso one is considering an F -inverse monoid T , it is well known [10] that, inthe L-representation of T , X is a semilattice and Y is a semilattice withmaximum. Our characterization of F -morphisms will generalize this.


For s ∈ S, we set sd = ss−1 and sr = s−1s. It is shown in [10] that, givena McAlister-O’Carroll triple (γ, q, Y ), there is a unique order-preserving sur-jection p : X → E(S) extending q such that:

(1) dom(eγ) = [e]p−1 for e ∈ E(S);(2) If xp = sr, then (sx)p = sd.

From now on we supress γ from the notation.We shall need the following lemma about McAlister-O’Carroll triples.

Lemma 6.11. Suppose (γ, q, Y ) is a McAlister-O’Carroll triple and x ∈ X.Then there exist y ∈ Y and s ∈ S such that yq = sr and sy = x.

Proof. By assumption, x = sy for some y ∈ Y and s ∈ S. Thus y ∈ dom(sr)whence yq ≤ sr. Then y ∈ dom(yq) so yqy = y. Hence x = sy = s(yq)y and(s(yq))r = yqsr(yq) = yq. So s(yq) is as desired. ¤Proposition 6.12. Let (γ, q, Y ) be a McAlister-O’Carroll triple and p beas above. Then the projection ϕ : L(γ, q, Y ) → S is an F -morphism if andonly if for each e ∈ E(S), pairs of elements of ep−1 have a meet, eq−1 hasa maximum element, and the function e 7→ max(eq−1) is order-preserving.

Proof. Suppose that the latter condition holds. We show that ϕ is an F -morphism. For e ∈ E(S), let ye = max(eq−1) and let s ∈ S. Then (sysr)p =sd so ysd ∧ sysr ∈ sdq−1 (recall: Y is an order ideal). Also

s−1(ysd ∧ sysr) ≤ s−1sysr = ysr

and hence is in Y . Thus (ysd ∧ sysr, s) ∈ L(γ, q, Y ). Suppose (y, s) ∈L(γ, q, Y ). We claim that y ≤ ysd ∧ sysr. Indeed, yq = sd so y ≤ ysd. Buts−1y ∈ srq−1 so s−1y ≤ ysr whence y ≤ sysr. It follows that (ysd ∧ sysr, s)is the maximum element of sϕ−1. Suppose s1 ≤ s2; then s1d ≤ s2d ands1r ≤ s2r, so ys1d ≤ ys2d and similarly ys1r ≤ ys2r. Thus

s1ys1r = s2ys1r ≤ s2ys2r

so (ys1d ∧ s1ys1r, s1) ≤ (ys2d ∧ s2ys2r, s2). Hence the projection is an F -morphism.

Suppose that the projection ϕ : L(γ, q, Y ) → S is an F -morphism. Toshow that pairs of elements of ep−1 have a meet, it suffices to show that,for x ∈ ep−1 and y ∈ eq−1, there exists x ∧ y. Indeed, suppose we haveshown this and that x, x′ ∈ ep−1. Then, by Lemma 6.11, x′ = sz for somez ∈ Y and s ∈ S such that zq = sr. So e = x′p = sd whence s−1x existsand (s−1x)p = sr. Therefore z ∧ s−1x and s(z ∧ s−1x) exist. But it is thenstraightforward to verify that x ∧ x′ = s(z ∧ s−1x).

So suppose x ∈ ep−1 and y ∈ eq−1. Let x = sz with z ∈ Y and zq = sr.Then e = (sz)p = sd. Let (k, s) = max(sϕ−1). Then kq = sd so y ∧ kexists. But (y ∧ k)q = sd, so s−1(y ∧ k) exists. Also, since s−1k ∈ Y and Yis an order ideal, s−1(y ∧ k) ∈ Y . Thus we can take s−1(y ∧ k) ∧ z which isin Y . It is then straightforward to check that q of this expression is sr. Solet h = s(s−1(y ∧ k) ∧ z). We claim h = x ∧ y. Indeed, h ≤ sz = x. Also


h ≤ ss−1y = y. Suppose that j ≤ x, y. Then jp ≤ e = sd so s−1j ≤ s−1x =z. Hence j, s−1j ∈ Y and jq ≤ e = sd. So (j, jqs) ∈ L(γ, q, Y ). Suppose(k′, jqs) = max((jqs)ϕ−1). Then k′ ≤ k and j ≤ k′, so j ≤ k. Putting thisall together, we see that s−1j ≤ s−1(y ∧ k) ∧ z so j ≤ h.

We now show that eq−1 has a maximum. Let (y, e) = max(eϕ−1). Sup-pose y′ ∈ eq−1. Then (y′, e) ∈ L(γ, q, Y ) so y′ ≤ y. It follows immediatelythat e 7→ max(eq−1) preserves order since e 7→ max(eϕ−1) preserves or-der. ¤

6.3. The Expansion. For each inverse semigroup S, we shall constructan inverse semigroup SPr together with an F -morphism ηS : SPr → Swhich will be the universal F -morphism to S. The construction S 7→ SPr isfunctorial and the surjective morphisms ηS for the components of a naturaltransformation from ( )Pr to the identity functor. It follows that ( )Pr is anexpansion of inverse semigroups.

We shall construct SPr by first constructing a McAlister-O’Carroll triplefrom S which satisfies Proposition 6.12 of Section 6.2. This will enable usto build SPr and ηS explicitly. We shall then prove that ηS is the requireduniversal F -morphism. The triple (γS , qS , YS) will be constructed in severalstages with various intermediary propositions which prove that things arewell defined.

This expansion will be similar to SSl, only we will take S as the generatingset and we shall be interested only in vertices of Schutzenberger graphsrather than subgraphs.

For those readers preferring a more direct construction, the reader isreferred to Proposition 6.16.

If A is a set, we use Pfin(A) for the collection of non-empty finite subsetsof A. Let

XS = {A | A ∈ Pfin(Re) some e ∈ E(S)}and

YS = {A ∈ XS | A ∩ E(S) 6= ∅}.Suppose A ∈ Pfin(Re) and B ∈ Pfin(Rf ). Define A ∧ B = fA ∪ eB. It iseasy to see that A ∧ B ∈ Pfin(Ref ). Also note that A ∧ A = eA ∪ eA = Aand A ∧B = B ∧A. So XS is a semilattice.

Proposition 6.13. Let A ∈ Pfin(Re) and B ∈ Pfin(Rf ). Then A ≤ B ifand only if e ≤ f and eB ⊆ A.

Proof. Suppose A ≤ B. Then

A = A ∧B = fA ∪ eB ∈ Pfin(Ref )

so e = ef and eB ⊆ A. For the converse,

A ∧B = fA ∪ eB = A ∪ eB = A.

¤


It is now easy to see that YS is an order ideal (and hence a semilattice inits induced order). Define γS : S → I(XS ,≤) as follows:

(1) dom(sγS) = {A ∈ XS | A ∈ Pfin(Re), e ≤ sr};(2) sγ(A) = {sa | a ∈ A}.

Again, we supress γ and merely write sA. This action is, in some sense,induced by the Preston-Wagner representation of S and the usual left actionof S on its power set. If S is a semigroup, s ∈ S, and X ⊆ S, then we willalso let sX = {sx | x ∈ X}. There should be no confusion since the aboveactions differ only in their domain of definition.

Proposition 6.14. The function γS is a well-defined homomorphism.

Proof. First note that dom(sγS) is an order ideal. Also, it is easy to see that

ran(sγS) = {A ∈ XS | A ∈ Pfin(Re), e ≤ sd}.Suppose A ≤ B ∈ dom(sγS), A ∈ Pfin(Re), B ∈ Pfin(Rf ), and e, f ≤ sr.Then e ≤ f and eB ⊆ A; so ses−1 ≤ sfs−1. Now ses−1sB = seB ⊆ sA.Thus sB ≤ sA. We note that s−1γS is the inverse morphism to sγS and sosγ is an isomorphism of order ideals of X.

Now

dom((st)γS) = {A ∈ XS | A ∈ Pfin(Re), e ≤ t−1srt}.But, for A ∈ dom(tγS), tA ∈ dom(sγS) if and only if A ∈ Pfin(Re) withe ≤ t−1t and tet−1 ≤ s−1s which is if and only if e = t−1tet−1t ≤ t−1srt. Inthis case (st)A = s(tA). ¤

Define qS : YS → E(S) by letting, for A ∈ Pfin(Re), AqS = e. Then qS

is a surjective homomorphism. Suppose A ∈ Pfin(Re) and let s ∈ A. Thensd = e and s−1A ∈ YS . If A ∈ YS and e ∈ E(S), then A ∈ dom(eγS) ifand only if Aq ≤ e. Finally, if s ∈ S, then Rsd ∈ YS , RsdqS = ss−1 ands−1Rsd ∈ YS .

We have thus proved the following.

Theorem 6.15. (γS , qS , YS) is a McAlister-O’Carroll triple.

Note that if A ∈ Pfin(Re), then ApS = e where pS is the extension of qS

to XS .We let SPr = L(γS , qS , YS) and ηS : SPr → S be the projection. Then,

by Proposition 6.12, ηS is an F -morphism. Indeed, since XS is a semilattice,we just need to show that eq−1

S has a maximum element for each e ∈ E(S)and that the function e 7→ max(eq−1

S ) preserves order. But clearly {e} issuch a maximum element and if e ≤ f , then {e} ≤ {f}. Finally, note thatif ψ : S → T is a homomorphism of inverse semigroups, then there is aninduced morphism ψ∗ : SPr → TPr sending (A, s) to (Aψ, sψ). Clearly η isa natural transformation to the identity functor.

We now wish to give a description of SPr in terms of the operation ofunion on Pfin(S) and the usual left action of S on this set.


Proposition 6.16. SPr = {(A, s) ∈ XS×S | sd, s ∈ A} with multiplication:

(A, s)(B, t) = (stt−1s−1A ∪ sB, st).

Proof. If (A, s) ∈ SPr, then A ∈ YS and AqS = sd, whence sd ∈ A. Also,since s−1A ∈ YS , s−1s ∈ s−1A and so s ∈ A. Conversely, if sd, s ∈ A, thenA, s−1A ∈ YS and Aq = sd. So (A, s) ∈ SPr. As for the multiplication,

(A, s)(B, t) = (s(s−1A ∧B), st)

by definition of multiplication in an L-semigroup, But

s−1A ∧B = tt−1s−1A ∪ s−1sB,

so s(s−1A ∧B) = stt−1s−1A ∪ sB. The result follows. ¤

If S is a group, then XS = Pfin(S) and the above result shows that SPr

is the usual prefix expansion of S. Note that if S is finite, then XS is finitewhence SPr is also finite.

It is not hard to see that SPr is a subsemigroup of the λ-semidirect prod-uct Pfin(S)oλ S.

All that remains is to show that ηS is universal.

Theorem 6.17. Let S be an inverse semigroup. Then the morphism ηS :SPr → S is universal amongst F -morphisms to S.

Proof. Let (γ, q, Y ) be a McAlister-O’Carroll triple as in Proposition 6.12.Let p be the extension of q to X. For e ∈ E(S), we let ye = max(eq−1).Since morphisms in the comma category of idempotent-pure morphisms overS correspond to morphisms (in the obvious sense) of McAlister-O’Carrolltriples, we just need to show that there is a unique morphism ψ : XS → Xpreserving the structure of a McAlister-O’Carroll triple, satisfying

(*) {sd, s}ψ = ysd ∧ sysr.

The induced morphism ψ : SPr → L(γ, q, Y ) defined by ψ× 1S will then bethe unique morphism preserving maxima.

Let A ∈ X, ApS = e. We define ψ by Aψ =∧

s∈A sysr. Since s ∈ A,sd = e and so (sysr)p = sd. Thus, by hypothesis on (γ, q, Y ), Aψ is welldefined. Also, ψq = qS (obvious) and YSψ ⊆ Y (since, for A ∈ YS , yq ∈ A,so Aψ ≤ yqyyq = yyq). Clearly, {sd, s}ψ = ysd ∧ sysr.

To see that ψ preserves order, suppose that A ≤ B with A ∈ Pfin(Re),B ∈ Pfin(Rf ). Then e ≤ f and eB ⊆ A. Observe that, for t ∈ B, (et)r =t−1et ≤ t−1t = tr, so y(et)r ≤ ytr whence

(et)y(et)r = ty(et)r ≤ tytr.

Therefore,

Aψ ≤∧

t∈B

(et)y(et)r ≤ Bψ.


Now we show that ψ|YSpreserves meets. Let A,B ∈ YS . Since ψ preserves

order, we need only show Aψ∧Bψ ≤ (A∧B)ψ. Suppose AqS = e, BqS = f .Now

(A ∧B)ψ = (fA ∪ eB)ψ =∧

s∈A

fsy(fs)r ∧∧

t∈B

ety(et)r.

On the other hand,

Aψ ∧Bψ =∧

s∈A

sysr ∧∧

t∈B

tytr =∧

s∈A

(sysr ∧ yf ) ∧∧

t∈B

(tytr ∧ ye);

the last equality follows because e ∈ A, f ∈ B. Thus, it suffices to showthat, for s ∈ A, sysr ∧ yf ≤ fsy(fs)r and dually for t ∈ B.

Since Y is an ideal, sysr ∧ yf ∈ Y . Also (sysr)p = e, so (sysr ∧ yf )q ≤ efwhence sysr ∧ yf ≤ yef . Since yef ≤ yf ,

sysr ∧ yf = sysr ∧ yf ∧ yef = sysr ∧ yef .

Now observe s−1(sysr ∧ yef ) ≤ ysr ∧ s−1yef . However, the latter is ysr ∧s−1fyef . But

(ysr ∧ s−1fyef )q ≤ sr(s−1f)d = s−1fs = (fs)r,

so ysr ∧ s−1yef ≤ y(fs)r. We conclude s−1(sysr ∧ yf ) ≤ y(fs)r whence sysr ∧yf ≤ sy(fs)r = fsy(fs)r as desired.

Finally, we show that ψ preserves the action. Suppose that t ∈ S andAp ≤ tr. Then, for s ∈ A, (ts)r = s−1t−1ts ≤ s−1s and s−1s = s−1ss−1s ≤s−1t−1ts = (ts)r (since ss−1 ≤ t−1t) whence (ts)r = sr. So

(tA)ψ =∧

s∈tA

sysr =∧

s∈A

tsy(ts)r =∧

s∈A

tsysr.

But it is easy to check that multiplication by t preserves finite meets (sinceit is an order isomorphism with inverse, multiplication by t−1) so the righthand side is t(Aψ).

To show that ψ is unique, suppose τ : XS → X induces another morphismof McAlister-O’Carroll triples satisfying (∗). First of all, since, for all A ∈XS , A = sB for some s ∈ S, B ∈ YS , and since τ preserves the action, itsuffices to show that τ agrees with ψ on YS . Let A = {e, s1, . . . , sn} ∈ YS

with AqS = e. Then

A = {e, s1} ∪ · · · ∪ {e, sn} = {e, s1} ∧ · · · ∧ {e, sn}.Since, for all i, {e, si}τ = ye ∧ siysir by (∗), and since τ |YS

preserves meets,it follow that Aτ =

∧s∈A(ye ∧ sysr) = Aψ (since e ∈ A). ¤

6.4. Structure. We now study the structure of SPr. The following resultsare straightforward deductions from the structure of L-semigroups.

Proposition 6.18. Let S be an inverse semigroup.(1) (A, s) ≤ (B, t) if and only if s ≤ t and ss−1B ⊆ A.(2) (A, s) R (B, t) if and only if A = B.(3) (A, s) L (B, t) if and only if s−1A = t−1B.


(4) Let (A, e) ∈ E(SPr) and H ′e = {s ∈ He | sA = A}. Then ηS induces

an isomorphism of H(A,e) with H ′e.

(5) (A, s) D (B, t) if and only if there exists u ∈ S such that ud = sr,u ∈ s−1A, and u−1s−1A = B.

Corollary 6.19. The R-classes of SPr are finite; hence SPr is residuallyfinite. Moreover, if the maximal subgroups of S are torsion-free, then SPr

is combinatorial.

Proof. The first assertion follow from Proposition 6.18 (2) since there areonly finitely many elements of SPr of the form (A, s) (since A is finite). Thelast assertion follows since the H-classes of SPr must then be finite and, byProposition 6.18 (4), are isomorphic to subgroups of the maximal subgroupsof S. ¤

We end this subsection by observing that SPr is generated by S via themap

s 7→ ({sd, s}, s).Indeed, this follows from the universal property since if T ⊆ SPr is theinverse subsemigroup generated by the elements of the form ({sd, s}, s),then the restriction ηS |T : T → S would be an F -morphism and wouldbe universal. Since the projection from SPr is idempotent-pure, there is anatural surjective morphism ψ : SSl → SPr (where we view S as generatedby S). This allows an alternate proof that if S is finite, then so is SPr. Thismap is easily seen to take a pair (X, s) to the pair (Ver(X), s) where Ver(X)is the set of vertices of X.

6.5. Applications. Our main application is to partial actions of inversesemigroups. A partial action of an inverse semigroup S on a set X consistsof an order-preserving dual prehomomorphism θ : S → I(X). These mostcommonly arise from a restricting an action of S by partial bijections of aset Y to a subset X ⊆ Y . We say the partial action is graded if there is amap p : X → E(S) such that: for each idempotent e ∈ E(S), ∃ex if andonly if xp ≤ e; if xp = s−1s, then (sx)p = ss−1. The first and third authorcan show that all graded partial actions arise from restricting actions to asubset; that is, extending the terminology of [9], graded partial actions canbe globalized. It is not apparent this is true of all partial actions of inversesemigroups.

For applications of partial group actions, see Exel’s [7] and the first authorand Kellendonk’s [9].

Proposition 6.20. Let S be an inverse semigroup and X a set. Then allpartial actions of S on X are of the form η∗Sϕ where ϕ : SPr → I(X) is anaction.

Proof. By the universal property of SPr and by Theorem 6.10, there existsa bijection between homomorphisms ϕ : SPr → I(X) and order-preservingdual prehomomorphisms θ : S → I(X) given by ϕ 7→ η∗Sϕ. ¤


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Division of Mathematics, School of Informatics, University of Wales, GwyneddLL57 1UT, Wales

E-mail address: [email protected]

Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, IsraelE-mail address: [email protected]

Department of Pure Mathematics, Faculty of Sciences, University of Porto,

4099-002 Porto, Portugal1

E-mail address: [email protected]

1Current Address: School of Mathematics and Statistics, Carleton University, 1125Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

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