The etale groupoid of an inverse semigroup as a groupoid of filters

THE ´ ET ALE GR OUPOID OF AN INVERSE SEMIGR OUP AS A GR OUPOID OF FIL TERS M. V. LA WSON, S. W. MA RGOLIS, AND B. STEINBERG Abstract. P aterson show ed how to construct an ´ etale groupoid from an in- v erse s emi group using i deas from functional analysis. This construction was later simplified b y Lenz. W e sho w that Lenz’s construction can itself be fur ther simplified b y usi ng filters: the top ological groupoi d asso ciated with an inv erse semigroup is precisely a group oid of filters. In addition, idemp oten t filters are closed i n verse subsemigroups and so determine transitive r epresen tations by means of partial bijections. This connection b et wee n filters and representa- tions by partial bijections is exploited to show how linear representat ions of inv erse semigroups can b e constructed from the groups o ccuring in the asso ci- ated topol ogical group oid. 2000 Mathematics Subject Classific ation : 20M18, 20M30, 18B40, 46L05. This pap er is dedicated to the memory of our friend and colleague Steve Haata ja 1. Introduction and motiv a tion In his influential b o ok , Renault [25] show ed how to c o nstruct C ∗ -algebra s from lo cally compact top o lo gical group oids. This can b e seen as a far-re a ching g en- eralization of b o th commutativ e C ∗ -algebra s and finite dimensio nal C ∗ -algebra s. F rom this p e rsp ective, lo c ally compact topolog ical group oids can be viewed as ‘non-commutativ e top o logical s pa ces’. Rena ult also show ed that in addition to group oids and C ∗ -algebra s, a thir d cla ss of structur es naturally intervenes: inv erse semigroups. Lo cal bisections of top olog ical group oids form inv erse semig roups and, conv ersely , inv erse semigroups can b e us ed to construct top olog ical gr oup oids. The relationship betw een in verse s emigroups a nd to po logical g roup oids can b e seen as a g eneralization of that betw een (pre)sheaves of g roups and their c orre- sp onding display spaces, s ince a n in verse semigroup with central idemp otents is a presheaf of gro ups over its semilattice o f idemp otents. This r elationship has b een inv estig ated by a num b er of authors: notably Paterson [23], Kellendonk [5, 6, 7, 8] and Resende [26]. Our pap er is r elated to Paterson’s work but mediated through a more r ecent redaction due to Daniel Lenz [17]. W e prov e tw o ma in results. First, w e sho w that Lenz’s construction of the top ological group oid can b e interpreted entirely in ter ms of down-directed cos ets on inv erse semigroups — these are pre cisely the filters in an inv erse semigroup. Such filters arise naturally from thos e transitive actions which we term ‘universal’. Second, we show how representations of an inv er se semigr o up can b e constructed from the gro ups o ccuring in the asso ciated topolo gical gr oup oid. This is rela ted to Steinberg’s results on constructing finite-dimensional representations of inv erse 1 2 M. V. LA WSON, S. W. M AR GOLIS, AND B. STEI NBER G semigroups using gr oup oid techniques descr ib ed in [34]. The first result prov ed in this pa per has alr e ady b een develop ed further in [13, 14]. Lenz [17] was the main spur that led us to write this pap er but in the cour se of doing so , we realized that the first four chapters of Ruyle’s unpublished thesis [27] could be viewed a s a ma jor contribution to the a ims of this pap er in the ca se of free inv erse monoids . Ruyle’s work has prov ed indispe ns ible for our Sec tio n 2. In addition, Leech [1 6], with its emphasis on the order -theoretic structure of inv erse semigroups, can be seen with mathematica l hindsigh t to be a precursor of our approach. Last, but not least, Boris Schein in a num ber of s e minars talked ab out wa y s o f constructing infin itesimal elements of an in verse semigro up: the maximal filters of an inv erse se mig roup ca n b e regar ded as just that [2 9, 30]. F or gener al inverse semigro up theory we refer the reader to [11]. How ever, we note the following. The pro duct in a semigro up will usua lly b e denoted b y c o n- catenation but s o metimes we s hall use · for emphasis; we shall a lso use it to deno te actions. In an inverse semigroup S we define d ( s ) = s − 1 s a nd r ( s ) = ss − 1 . Green’s relation H can b e defined in terms of this notation a s follows: s H t if and only if d ( s ) = d ( t ) a nd r ( s ) = r ( t ). If e is an idemp otent in a semigroup S then G e will denote the H -cla ss in S con taining e ; this is a maximal subgr oup. The natural par tial o r der will b e the o nly par tial order considere d when w e dea l with inv er se semigr oups. If X ⊆ S then E ( X ) denotes the set of idemp otents in X . An inv er se subs e migroup of S is sa id to be wide if it contains all the idemp otents of S . A primitive idemp otent e in an inverse s e migroup S with zero is one with the prop erty that if f ≤ e then either f = e or f = 0 . Let S b e an inv e rse semigro up. The minimum gr oup c ongru enc e σ on S is defined by a σ b iff c ≤ a, b for some c ∈ S . This co ngruence ha s the pr op erty that S /σ is a g roup, and if ρ is any congr uence on S for which S/ρ is a group, we hav e that σ ⊆ ρ . W e denote by σ ♮ the a sso ciated natural ho momorphism S → S/ σ . See [11] for more information o n this imp orta n t congruence. 2. The structure of transitive a ctions In this s ection, we shall b egin by reviewing the general theo ry o f repres ent ations of inv erse semigr oups by partial p ermutations. C ha pter IV, Sectio n 4 of [24] co n- tains an exp os itio n of this elementary theory and we refer the rea der there for any pro ofs we omit. W e also incorp or ate some results by Ruyle from [27] which can be viewed as ant icipating so me of the idea s in this pap er. W e then in tro duce the concept of universal transitive a c tio ns which provides the co nnection with the work of L e nz to b e explained in Section 3. 2.1. The classical theo ry. A re pr esentation of an inverse se migroup b y mea ns of partial bijectio ns (or par tial p er m utations) is a homo morphism θ : S → I ( X ) to the symmetric inv erse monoid on a set X . A representation of an inv erse semigroup in this sens e leads to a cor resp onding notion of a n actio n of the in verse semigr oup S on the set X : the as s o ciated a ction is defined by s · x = θ ( s )( x ), if x belo ngs to the set-theoretic doma in o f θ ( s ). The action is therefore a partial function fro m S × X to X mapping ( s, x ) to s · x when ∃ s · x sa tisfying the tw o a xioms: (A1): If ∃ e · x whe r e e is a n idemp otent then e · x = x . (A2): ∃ ( st ) · x iff ∃ s · ( t · x ) in which cas e they ar e e qual. INVERSE SEMIGROUPS AND GR OUPOIDS 3 It is ea sy to chec k that representations and a ctions are different wa ys of descr ibing the same thing. F o r co nv enience , we shall use the words ‘action’ and ‘represe ntation’ int erchangeably: if we say the in verse semigro up S acts on a set X then this will imply the ex istence of an appropr iate homomor phism from S to I ( X ). If S a c ts o n X we sha ll o ften refer to X as a sp ac e or as an S -sp ac e and its elements as p oints . A subset Y ⊆ X closed under the action is called a subsp ac e . Disjoint unions of actions a r e aga in actions. An action is said to b e effe ctive if for ea ch x ∈ X ther e is s ∈ S suc h that ∃ s · x . W e shall assume that all our actio ns are e ffectiv e. An effective actio n o f an inv erse s emigroup S on the set X induces an eq uiv alence relation ∼ on the set X when we define x ∼ y iff s · x = y for some s ∈ S . The action is said to b e tr ansitive if ∼ is X × X . Just a s in the theor y of p ermutation representations of gro ups, every representation o f an in verse semigroup is a disjoint union of trans itive re presentations. Thus the trans itiv e r epresentations o f inv erse semigroups are of esp ecial significa nce. Let X a nd Y be S -spaces. A morphism fro m X to Y is a function α : X → Y such that ∃ s · x implies that ∃ s · α ( x ) a nd α ( s · x ) = s · α ( x ). A st r ong morphism from X to Y is a function α : X → Y such that ∃ s · x ⇔ ∃ s · α ( x ) and if ∃ s · x then α ( s · x ) = s · α ( x ). Bijectiv e strong mo r phisms are ca lled e quivalenc es . The pro ofs of the following tw o lemmas are s traightforw ard. Lemma 2. 1. (i): Identity functions ar e (s t r ong) morphisms. (ii): The c omp osition of (st r ong) morphisms is again a (stro ng) m orphism. Lemma 2. 2. L et S b e an inverse semigr oup acting on X , Y and Z (i): The image of a s t r ong morphism α : X → Y is a subsp ac e of Y . (ii): If X and Y ar e tr ansitive S -sp ac es and α : X → Y is a st ro ng morphism then α is surje ctive. If we fix a n inv erse semigr o up S there are a num b er of ca tegories of actions a s- so ciated with it: actions a nd mo r phisms, actio ns and strong morphisms, tr ansitive actions and mor phisms, and tr a nsitive actions and strong mo rphisms. As we indi- cated above, these tw o catego ries of transitive a ctions will b e of central imp orta nce. A c ongruenc e on X is an eq uiv alence relatio n ∼ on the set X such that if x ∼ y and if ∃ s · x and ∃ s · y then s · x ∼ s · y . A str ong c ongruen c e on X is a n equiv alence relation ≈ o n the set X s uch that if x ≈ y and s ∈ S we have that ∃ s · x ⇔ ∃ s · y , and if the actions ar e defined then s · x ≈ s · y . Strong morphisms and s trong congruences a re united by a cla ssical first iso mor- phism theorem. Recall that the kernel of a function is the equiv a le nce r elation induced on its domain. The pro ofs of the following are r outine. Prop ositio n 2.3. (i): L et α : X → Y b e a st r ong morphism. Then the kernel of α is a stro ng c ongruenc e. (ii): L et ∼ b e a str ong c ongruen c e on X . Denote the ∼ -class c ontaining the element x by [ x ] . Define s · [ x ] = [ s · x ] if ∃ s · x . Then this defines an action S on the set of ∼ -c ongruenc e classes X/ ∼ and the natur al map ν : X → X/ ∼ is a st r ong morphism. (iii): L et α : X → Y b e a str ong morphism, let its kernel b e ∼ and let ν : X → X/ ∼ b e the asso ciate d n atur al map. Then t her e is a unique inje ctive st r ong morphism β : X/ ∼→ Y such that β ν = α . 4 M. V. LA WSON, S. W. M AR GOLIS, AND B. STEI NBER G The ab ov e result tells us that the category of tra ns itive repr esentations of a fixed inv er se s emigroup with stro ng morphisms b et ween them has a pa r ticularly nice s tructure. W e may analyze tr ansitive actions o f inv erse semigroups in a w ay genera lizing the re la tionship b etw een tr ansitive group actions a nd subgr oups. T o describ e this relationship we need some definitions. If A ⊆ S is a subs e t then define A ↑ = { s ∈ S : a ≤ s for some a ∈ A } . If A = A ↑ then A is said to b e close d (upwar ds) . Let X b e an S -space. Fix a p oint x ∈ X , a nd cons ider the set S x consisting of all s ∈ S such that s · x = x . W e call S x the stabilizer of the p oint x . Remark 2.4. W e do not assume in this pap er that ho momorphisms of inv erse semigroups with zer o preserve the zero. If θ : S → I ( X ) is a r epresentation that do es preser ve zero then the zero o f S is mapp ed to the empt y function of I ( X ). Clearly , the empty function cannot b elong to any stabilizer. W e s ay that a closed inv er se subsemigroup is pr op er if it does not contain a zero. In the theory w e summarize below, prop er closed in verse subsemigr oups ar ise fro m a ctions wher e the zero acts as the empty partial function. Now le t y ∈ X b e any p oint. By transitivity , ther e is an e le men t s ∈ S such that s · x = y . Observe that bec a use s · x is defined so to o is s − 1 s a nd that s − 1 s ∈ S x . The s et of all elements of S whic h map x to y is ( sS x ) ↑ . Let H be a closed inv erse subs e migroup of S . Define a left c oset of H to b e a set of the form ( sH ) ↑ where s − 1 s ∈ H . W e give the pro of of the following for completeness. Lemma 2. 5. (i): Two c osets ( sH ) ↑ and ( tH ) ↑ ar e e qual iff s − 1 t ∈ H . (ii): If ( sH ) ↑ ∩ ( tH ) ↑ 6 = ∅ then ( sH ) ↑ = ( tH ) ↑ . Pr o of. (i) Supp ose that ( sH ) ↑ = ( tH ) ↑ . Then t ∈ ( sH ) ↑ and so sh ≤ t for some h ∈ H . Thus s − 1 sh ≤ s − 1 t . B ut s − 1 sh ∈ H and H is closed and so s − 1 t ∈ H . Conv ersely , suppo se that s − 1 t ∈ H . Then s − 1 t = h for some h ∈ H and so sh = ss − 1 t ≤ t . It follows that tH ⊆ sH a nd so ( tH ) ↑ ⊆ ( sH ) ↑ . The reverse inclusion follows from the fact that t − 1 s ∈ H since H is closed under inv erses. (ii) Supp ose that a ∈ ( sH ) ↑ ∩ ( tH ) ↑ . Then s h 1 ≤ a a nd th 2 ≤ a for so me h 1 , h 2 ∈ H . Thus s − 1 sh 1 ≤ s − 1 a and t − 1 th 2 ≤ t − 1 a . Hence s − 1 a, t − 1 a ∈ H . It follo ws that s − 1 aa − 1 t ∈ H , but s − 1 aa − 1 t ≤ s − 1 t . This gives the result by (i) ab ov e.  W e denote by S/H the s et o f all left cosets of H in S . The inv erse semig r oup S acts on the set S/H when we define a · ( sH ) ↑ = ( asH ) ↑ whenever d ( as ) ∈ H . This defines a transitive action. The following is Lemma IV.4.9 of [24] a nd P rop o- sition 5.8.5 of [4]. Theorem 2.6. L et S act tr ansit ively on the set X . Then the action is e quivalent to the action of S on the set S/S x wher e x is any p oint of X . The following is Pr op osition IV.4.13 o f [24]. INVERSE SEMIGROUPS AND GR OUPOIDS 5 Prop ositio n 2.7 . If H and K ar e any close d inverse subsemigr oups of S then they determine e quivalent actions if and only if ther e exists s ∈ S such that sH s − 1 ⊆ K and s − 1 K s ⊆ H . The ab ove relationship b etw een closed inv erse subsemigro ups is called c onjugacy and defines an equiv alence relatio n on the set of closed inv er se subsemigr oups. The pro of of the following is given fo r completeness. Lemma 2. 8. H and K ar e c onjugate if and only if ( sH s − 1 ) ↑ = K and ( s − 1 K s ) ↑ = H . Pr o of. Let H and K b e conjugate. Let e ∈ H b e any idemp otent. Then ses − 1 ∈ K . But ses − 1 ≤ ss − 1 and so ss − 1 ∈ K . Similar ly s − 1 s ∈ H . W e hav e that sH s − 1 ⊆ K and so ( sH s − 1 ) ↑ ⊆ K . Let k ∈ K . Then s − 1 k s ∈ H and s ( s − 1 k s ) s − 1 ∈ sH s − 1 and s ( s − 1 k s ) s − 1 ≤ k . Thus ( sH s − 1 ) ↑ = K , as require d. The converse is immediate.  Thu s to study the tra ns itive a ctions of an inv er se semigr oups S it is enough to study the closed inv erse subsemig roups of S up to conjuga cy . The following result is mo tiv ated by Lemma 2 .1 6 of Ruyle’s thesis [27] a nd brings morphisms a nd strong morphisms back into the picture. Theorem 2.9. L et S b e an inverse semigr oup acting tr ansitively on the sets X and Y , and let x ∈ X and y ∈ Y . L et S x and S y b e the stabilizers in S of x and y r esp e ctively. (i): Ther e is a (unique) morphism α : X → Y such that α ( x ) = y iff S x ⊆ S y . (ii): Ther e is a (un ique) str ong morphism α : X → Y su ch t hat α ( x ) = y iff S x ⊆ S y and E ( S x ) = E ( S y ) . Pr o of. (i) W e b egin by proving uniqueness . Le t α, β : X → Y b e mo rphisms such that α ( x ) = β ( x ) = y . Let x ′ ∈ X be ar bitrary . By transitivity ther e exis ts a ∈ S such that x ′ = a · x . By the definition o f mo rphisms we hav e that ∃ a · α ( x ) and ∃ a · β ( x ) and that α ( x ′ ) = α ( a · x ) = a · α ( x ) and β ( x ′ ) = β ( a · x ) = a · β ( x ) . But by assumption α ( x ) = β ( x ) = y and so α ( x ′ ) = β ( x ′ ). It follows tha t α = β . Let α : X → Y be a morphism such that α ( x ) = y . Let s ∈ S x . Then ∃ s · x and s · x = x . By the definition o f morphism, it follows that ∃ s · α ( x ) and that α ( s · x ) = s · α ( x ). But s · x = x and so α ( x ) = s · α ( x ). Hence s · y = y . W e have therefore prov ed that s ∈ S y , and so S x ⊆ S y . Suppo se now that S x ⊆ S y . W e have to define a morphis m α : X → Y such tha t α ( x ) = y . W e start by defining α ( x ) = y . Let x ′ ∈ X b e any p oint in X . Then x ′ = a · x for some a ∈ S . W e nee d to s how that a · y exists. Since a · x exists we know that a − 1 a · x exists and this is equal to x . It follows that a − 1 a ∈ S x and s o a − 1 a ∈ S y , by assumption. Thus a − 1 a · y ex is ts and is equal to y . But from the existence of a − 1 a · y we ca n deduce the existence of a · y . W e would therefor e like to define α ( x ′ ) = a · y . W e hav e to chec k that this is w ell-defined. Supp ose that x ′ = a · x = b · x . Then b − 1 a · x = x and so b − 1 a ∈ S x . By a ssumption, b − 1 a ∈ S y and so b − 1 a · y = y . Thus bb − 1 a · y = b · y and bb − 1 a · y = bb − 1 · ( a · y ) = a · y . Thus a · y = b · y . It follows that α is a well-defined function mapping x to y . It r emains 6 M. V. LA WSON, S. W. M AR GOLIS, AND B. STEI NBER G to show that α is a morphism. Suppose that s · x ′ is defined. By a ssumption, there exists a ∈ S such that x ′ = a · x . By definition α ( x ′ ) = a · y . W e have that s · x ′ = s · ( a · x ) = sa · x . By definition α ( s · x ′ ) = sa · y . But sa · y = s · ( a · y ) = s · α ( x ′ ). Hence α ( s · x ′ ) = s · α ( x ′ ), as require d. (ii) W e b egin by proving uniqueness. Let α, β : X → Y be stro ng mo r phisms such that α ( x ) = β ( x ) = y . Let x ′ ∈ X b e ar bitrary . By transitivity there exists a ∈ S such that x ′ = a · x . By the definition of strong mor phisms w e hav e that ∃ a · α ( x ) a nd ∃ a · β ( x ) a nd that α ( x ′ ) = α ( a · x ) = a · α ( x ) and β ( x ′ ) = β ( a · x ) = a · β ( x ) . But by assumption α ( x ) = β ( x ) = y and so α ( x ′ ) = β ( x ′ ). It follows tha t α = β . Next we prove existence. Suppos e that S x ⊆ S y and E ( S x ) = E ( S y ). W e hav e to define a strong mor phism α : X → Y suc h that α ( x ) = y . W e star t by defining α ( x ) = y . Let x ′ ∈ X b e any point in X . Then x ′ = a · x for some a ∈ S . W e need to show that a · y exists. Since a · x exis ts we know that a − 1 a · x exists and this is equal to x . It follows that a − 1 a ∈ S x and so a − 1 a ∈ S y , by assumption. Thu s a − 1 a · y exists a nd is eq ual to y . But fro m the existence o f a − 1 a · y we ca n deduce the ex istence of a · y . W e therefore define α ( x ′ ) = a · y . W e have to chec k that this is well-defined. Supp o se that x ′ = a · x = b · x . Then b − 1 a · x = x a nd so b − 1 a ∈ S x . By assumption, b − 1 a ∈ S y and so b − 1 a · y = y . Thus bb − 1 a · y = b · y and bb − 1 a · y = bb − 1 · ( a · y ) = a · y . Thus a · y = b · y . It follows that α is a well-defined function mapping x to y . It remains to show that α is a s trong mor phism. Suppose that s · x ′ is defined. By assumption, there exists a ∈ S such that x ′ = a · x . By definitio n α ( x ′ ) = a · y . W e have that s · x ′ = s · ( a · x ) = sa · x . By definition α ( s · x ′ ) = sa · y . But sa · y = s · ( a · y ) = s · α ( x ′ ). Hence α ( s · x ′ ) = s · α ( x ′ ). Now s upp os e that α ( x ′ ) = y ′ and ∃ s · y ′ . W e shall prov e that ∃ s · x ′ . O bserve that ∃ s − 1 s · y ′ and that it is enough to prov e that ∃ s − 1 s · x ′ . Let x ′ = u · x , which exists since we ar e a ssuming that our action is tr a nsitive. The n by what we prov ed ab ov e we have that y ′ = u · y . Obser ve that u − 1 ( s − 1 s ) u · y = y and so u − 1 ( s − 1 s ) u ∈ E ( S y ). It fo llows by our assumption that u − 1 ( s − 1 s ) u ∈ E ( S x ) and so u − 1 ( s − 1 s ) u · x = x . It readily follows tha t ∃ s − 1 s · x ′ , a nd so ∃ s · x ′ , a s req uir ed. W e now prove the conv erse. Let α : X → Y b e a strong mor phism suc h that α ( x ) = y . Let s ∈ S x . Then ∃ s · x and s · x = x . By the definition o f strong morphism, it follows that ∃ s · α ( x ) and that α ( s · x ) = s · α ( x ). But s · x = x and so α ( x ) = s · α ( x ). Hence s · y = y . W e hav e therefore pr ov ed that s ∈ S y , and so S x ⊆ S y . Let e ∈ E ( S y ). Then ∃ e · α ( x ). But α is a str ong mo rphism and so ∃ e · x . Clearly e ∈ E ( S x ). It follows that E ( S x ) = E ( S y ).  The following result is adapted from Lemma 1.9 of Ruyle [2 7] and will b e useful to us later. Lemma 2.10. L et F b e a close d inverse subsemigr oup of the semilattic e of idem- p otents of the inverse su bsemigr oup S . Defin e F = { s ∈ S : s − 1 F s ⊆ F , sF s − 1 ⊆ F } . INVERSE SEMIGROUPS AND GR OUPOIDS 7 Then F is a close d inverse subsemigr oup of S whose semilattic e of idemp otents is F . F u rthermor e, if T is any close d su bsemigr oup of S with semilattic e of idemp otents F then T ⊆ F . Pr o of. Clearly the set F is clos ed under inverses. Le t s, t ∈ F . W e calcula te ( st ) − 1 F ( st ) = t − 1 ( s − 1 F s ) t ⊆ t − 1 F t ⊆ F and ( st ) F ( st ) − 1 = s ( tF t − 1 ) s − 1 ⊆ sF s − 1 ⊆ F . Thu s st ∈ F . It follows that F is an inv er se subsemig roup o f S . Let e ∈ F a nd f ∈ F . Then by assumption ef ∈ F . But e f ≤ e and F is a closed inv erse subsemigr oup of the semilattice o f idemp otents and so e ∈ F . Th us E ( F ) = F . Let s ≤ t where s ∈ F . Then s = s s − 1 t = f t . Let e ∈ F . Then s − 1 es = t − 1 f ef t = t − 1 ef t ≤ t − 1 et. Now s − 1 es, t − 1 et are idempotents and s − 1 es ∈ F thus t − 1 et ∈ F , beca use F is a closed inv er se subsemigroup of the semilattice of idemp otents. Similarly te t − 1 ∈ F . It follows that t ∈ F a nd so F is a closed inv erse s ubs e migroup o f S . Finally , let T b e a closed inv er se subsemigroup o f S suc h that E ( T ) = F . Let t ∈ T . Then fo r each e ∈ F we hav e that t − 1 et, tet − 1 ∈ F . Thu s T ⊆ F .  A closed inv erse subse mig roup T of S will b e said to b e ful ly close d if T = E ( T ). Clo s ed inv erse subsemigro ups of the semilattice of idemp otents of an inv erse semigroup a re called filter s in E ( S ). Obser ve the emphas is o n the word ‘in’. A filter in E ( S ) is said to b e princip al if it is o f the form e ↑ . W e deno te by F E ( S ) the s et o f all close d inv erse s ubsemigroups o f E ( S ) and ca ll it the filter sp ac e of the semilattic e of idemp otents of S . This filter spa ce is a p o set when we define F ≤ F ′ iff F ′ ⊆ F so that, in particular , e ↑ ≤ f ↑ iff e ≤ f . Let F b e a filter in E ( S ). Then F ↑ is a clo s ed in verse subse migroup contain- ing F and clea r ly the smallest such in verse subsemig r oup. O n the other hand, by Lemma 2.10 , F is the larg est closed in verse s ubsemigroup with semila ttice of idempo ten ts F . W e hav e ther efore pr ov ed the following. Lemma 2.11. The semilattic e of idemp otents of any close d inverse subsemigr oup H of an inverse semigr oup S is a filter F in E ( S ) and F ↑ ⊆ H ⊆ F . Thus F ↑ is the smal lest close d inverse subsemigr ou p with semilattic e of idemp otent s F and F is the lar gest. Prop ositio n 2. 1 2. L et S b e an inverse semigr oup and let G = S/σ . Then ther e is an inclusion-pr eserving bije ction b etwe en the wide close d inverse s ubsemigr oups of S and the su b gr oups of G . Pr o of. Let E ( S ) ⊆ T ⊆ S b e a wide inv erse subse mig roup. Then the image of T in G is a subgroup since inv erse subsemigro ups map to inv erse s ubsemigroups under homomorphisms. Suppose T and T ′ , where also E ( S ) ⊆ T ′ ⊆ S , hav e the same image in G . Let t ∈ T . Then σ ♮ ( t ) = σ ♮ ( t ′ ) for some t ′ ∈ T ′ . Th us a ≤ t, t ′ from the definition of σ . But b oth T and T ′ are order ideals o f S a nd so a ∈ T ∩ T ′ . Thu s a ≤ t and a ∈ T ′ and T ′ is clo sed thus t ∈ T ′ . W e hav e shown tha t T ⊆ T ′ . The re verse inclusion follows by symmetry . If H is a subgr oup of G then the full inv er se image of H under σ ♮ is a wide inv erse subsemig roup of S . This defines a n 8 M. V. LA WSON, S. W. M AR GOLIS, AND B. STEI NBER G order-pr eserving map going in the opp osite directio n. It is now clear that the re s ult holds.  The following is a sp ecial case o f Lemma 2.1 7 of [27]. W e include it for interest since we shall not use it explicitly . Lemma 2. 13. L et F b e a filter in E ( S ) in the inverse semigr oup S . (i): The interse ction of any family of close d inverse subsemigr oups with c om- mon semilattic e of idemp otents F is again a close d inverse su bsemigr oup with semilattic e of idemp otents F . (ii): Given any family of close d inverse subsemigr oups with c ommon semilat- tic e of idemp otents F ther e is a smal lest close d inverse subsemigr oup with semilattic e F which c ontains them al l. 2.2. Univ ersal and fundamen tal transitiv e actions . W e shall now define tw o sp ecial classes of transitive a ctions that play a decisive role in this pap er. Let S be a n inv er se semig roup and let H be a closed inv erse subsemigroup of S . By Lemma 2.10, we hav e that E ( H ) ↑ ⊆ H ⊆ E ( H ) where E ( H ) is a filter in E ( S ). W e shall use this observ atio n as the basis o f tw o definitions, the first of which is by far the most imp orta n t. W e sha ll say that a transitive S -space X is universal if the stabilizer of a p oint of X is the clo sure F ↑ for some filter F of E ( S ), and fundamental if the stabilizer of a po int of X is F for some filter F in E ( S ). Both definitions are indep endent o f the p oint chosen. Lemma 2. 14. (1) A str ong morphism b etwe en un iversal tr ansitive actions is an e quivalenc e. (2) A ny str ong morphism with domain a fundamental tr ansitive action and c o domain a tr ansitive action is an e quivalenc e. Pr o of. (1) Let X a nd Y be universal transitive s paces. Let α : X → Y be a str ong morphism. Cho ose x ∈ X . Then S x ⊆ S α ( x ) and E ( S x ) = E ( S α ( x ) ). But the actions a re universal and so all stabilizer s are the full clos ur es of their semilattices of ide mp otents. Thus S x = S α ( x ) and so α is an equiv alence by Theorem 2.9(ii). (2) Let X and Y b e tr ansitive s paces where X is fundamental a nd let α : X → Y be a str o ng morphism. Cho ose x ∈ X and let y = α ( x ). Then S x ⊆ S y and E ( S x ) = E ( S y ) b y Theor em 2.9 (ii). But S x is fundamental and so S x = S y . W e may deduce from Theore m 2.9 (ii) that there is a unique stro ng morphism from Y to X mapping y to x . It follows that α is a n eq uiv alence.  If α : X → Y is a stro ng morphism b etw een t wo tra nsitive S -spaces, we sha ll say that Y is str ongly c over e d b y X . The imp o rtance of universal actions ar ises from the following result. Prop ositio n 2.15. L et S b e an inverse semigr oup. (1) Each tra nsitive action of S is str ongly c over e d by a u niversal one. (2) Each tra nsitive action of S str ongly c overs a fun damental one. Pr o of. (1) Let Y b e a n arbitrar y transitive S - space. C ho ose a p oint y ∈ Y . Let F = E ( S y ) and put H = F ↑ . Then E ( H ) = E ( S y ) and H ⊆ S y . Put X = S/ H INVERSE SEMIGROUPS AND GR OUPOIDS 9 and choose the p oint x in X to be the c o set H . T hen ther e is a uniq ue strong morphism α : X → Y suc h that α ( x ) = y by Theor em 2.9(ii) which is s urjective b y Lemma 2.2(ii) and X is a universal tra nsitive spa ce b y constr uction. (2) Let Y b e an arbitrar y transitive S -space. Cho ose a po in t y ∈ Y . Let F = E ( S y ) and put H = F . Thus by Lemma 2.10 we hav e that S y ⊆ H a nd E ( S y ) = E ( H ). Put X = S/H a nd choose the p oint x in X to b e the coset H . Then there is a unique strong mor phism α : Y → X suc h that α ( y ) = x by Theo - rem 2.9 (ii) which is surjective by Lemma 2.2(ii) and X is a fundamental transitive space by constructio n.  Theorem 2.16. L et X b e a u niversal, tr ansitive S -sp ac e and let x b e a p oint of X . Put S x = F ↑ , wher e F is a filter in E ( S ) and G F = F /σ . Then ther e is an or der-pr eserving bije ction b etwe en t he set of str ong c ongruenc es on X and the set of su b gr oups of G F . Pr o of. Put G = G F . By Pro p o sition 2.12, there is an or der-preser ving bijectio n betw een the closed in verse subsemigr oups H suc h that F ↑ ⊆ H ⊆ F and the subgroups of G . Thu s we need to show that there is a bijection b et ween the set of stro ng congruences on X and the set of closed wide inv erse subsemigro ups of F . Observe that we use the fact that strong morphis ms b etw een transitive spa ces are surjective by Lemma 2.2(ii). Let ∼ b e a strong cong ruence defined on X . Then by P rop osition 2.3 it de- termines a strong morphism ν : X → X/ ∼ . F or x given in the s tatement of the theorem, we hav e that the stabilizer o f [ x ], the ∼ -class co n taining x , is a clo sed inv er se subsemigroup H x such that F ↑ ⊆ H x ⊆ F by Theo rem 2.9(ii). W e hav e th us defined a function fro m strong co ngruences on X to the set of clo sed wide inv er se subsemig roups o f F . Suppo se that ∼ 1 and ∼ 2 are tw o strong co ngruences on X that map to the same closed wide in verse subsemigr oup. Denote the ∼ i equiv alence cla ss co nt aining x by [ x ] i and let ν i : X → X/ ∼ i be the natural map. Let x ∈ X . Then the stabilizer of [ x ] 1 and the stabilizer of [ x ] 2 are the sa me: na mely H . Suppose that x ∼ 1 y . Thu s [ x ] 1 = [ y ] 1 . Since X is an universal transitive S -space there is b ∈ B such that b · x = y . It follows that b · [ x ] 1 = [ y ] 1 = [ x ] 1 and so b ∈ H . By assumption b · [ x ] 2 = [ x ] 2 . But ∼ 2 is a stro ng congruenc e and s o y = b · x ∼ 2 x and so x ∼ 2 y . A symmetrical a rgument shows that ∼ 1 and ∼ 2 are eq ua l. T hus the corresp o ndence we hav e defined is injective. W e now show that it is surjective. Let F ↑ ⊆ H ⊆ F b e such a clos e d wide inv erse subsemigr oup. The n Y = H /S is a tra nsitive S -space. Cho o se the p oint y = H ∈ Y . Then by T heo rem 2 .9 (ii) there is a unique stro ng morphism α H : X → Y such that α ( x ) = y . The kernel of α H , which we denote by ∼ H , is a str ong congruence defined on X by P r op osition 2 .3, and the kernel of α H maps to H .  Observe that the a b ove theor em r equires a chosen p oint in X . 2.3. A top olo g ical in terpretation. Let S b e an in verse s emigroup and X an S -space. Define an S -lab eled graph G ( X ) whose vertices are X and who se edges go from x to sx , where x ∈ X , s ∈ S and sx is defined, with label s o n this edge in this case. There is an o bvious involution on the graph by inv ersion, s o this is a graph in the sense of Serr e . Observe that the directed gr aph G ( X ) is c o nnected iff 10 M. V. LA WSON, S. W. M AR GOLIS, AND B. STEI NBER G X is trans itive. F rom now on we shall dea l only with tr anstive a c tions and s o our graphs will b e connected. The star of a vertex x in G ( X ) is the set o f all edges that start a t x . Now let G a nd H b e a rbitrary g raphs. A morphism f from G to H is c alled an immersion if it induces an injection from the star set of x to that of f ( x ) for each vertex x of G . The mo rphism f is called a c over if it induces a bijection b etw een such star sets. The following is the key link be tw een the algebra ic and the top ologica l int erpretatio ns of inv erse semigroup actions. Lemma 2. 1 7. L et S b e an inverse semigr oup and let X and Y b e tr ansitive S - sp ac es. Ther e is a morphism fr om X to Y iff ther e is a lab el pr eserving immersion fr om G ( X ) t o G ( Y ) , and ther e is a str ong morphism fr om X to Y iff ther e is a lab el pr eserving c over fr om G ( X ) to G ( Y ) . Pr o of. Let α : X → Y b e a mor phism of transitive S -spaces. Consider the directed edge x s → y in the graph G ( X ). Then s · x = y . Since α is a morphism, w e have that α ( s · x ) = s · α ( x ) = α ( y ). W e may therefore define f : G ( X ) → G ( Y ) by mapping the edge x s → y to the edge α ( x ) s → α ( y ). It is immediate that this is an immersion. The fa ct tha t immersions a rise fr om morphisms is now straightforw ard to prov e. Finally , supp ose that α is a strong morphism. Let α ( x ) s → α ( y ) be an edge. This means that s · α ( x ) = α ( y ). But α is a stro ng morphism and so s · x exists and α ( s · x ) = s · α ( x ). It follows tha t the graph map is a cov er.  F or a mor e complete account of the connection be t ween immersions, inverse monoids and inv e rse ca tegories see [20, 33]. 3. The ´ et al e groupoid asso cia ted with an inverse semigroup In Section 2, we inv estigated the relationship b etw een tra nsitive a c tions of an inv er se semigr oup and clos ed inv erse subsemigroups. W e found that the universal transitive actions played a s pec ia l role. W e shall show in this section how these universal transitive actions, via their stabilizer s, lea d to the inv erse semigro up in- tro duced by Lenz and thence to Paterson’s ´ etale gr oup oid. 3.1. The inv erse semigroup of cose ts K ( S ) . W e beg in by reviewing a construc- tion s tudied by a num b er of author s [3 1, 16, 10, 1 1]. A subs e t A ⊆ S of an in verse semigroup is called an atlas if A = AA − 1 A . A clo sed a tlas is precisely a c o set o f a closed inv erse subsemigro up of S [10]. W e shall therefore refer to a closed atlas as a c oset . O bs erve that the intersection of co sets, if non-empty , is a co set. The set of cosets of S is deno ted by K ( S ). There is a pro duct o n K ( S ), denoted by ⊗ , and defined as follows: if A, B ∈ K ( S ) then A ⊗ B is the intersection o f all cosets of S that contain the set AB . More ex plicitly if X = ( aH ) ↑ , where a − 1 a ∈ H , and Y = ( bK ) ↑ , where b − 1 b ∈ K , then X ⊗ Y = ( ab h b − 1 H b, K i ) ↑ where h C, D i is the inv erse s ubsemigroup of S g e nerated by C ∪ D . In fact, K ( S ) is an inv e r se semigroup called the (ful l) c oset semigr oup of S . No te that its natura l pa r tial order is reverse inclusion. Th us S is the zer o element of K ( S ). The idemp otents of K ( S ) are just the closed inv erse subs emigroups of S . There is an embedding ι : S → K ( S ) that maps s to s ↑ . Obs e rve now that if A ∈ K ( S ) then for ea ch s ∈ A we have that s ↑ ⊆ A a nd s o A ≤ s ↑ . It follows readily from this that A is in fact the meet of the s et { s ↑ : s ∈ A } . More genera lly , every non-empty subset o f K ( S ) has a meet and so the inv erse semigr oup K ( S ) INVERSE SEMIGROUPS AND GR OUPOIDS 11 is me et c omplete . The map ι : S → K ( S ) is universal for ma ps to meet complete inv er se semigroups. Thus the inv erse semigroup K ( S ) is the me et c ompletio n o f the inv er se semigroup S [16]. It is worth noting that the catego ry of meet co mplete inv er se semigroups a nd their morphisms is not a full sub ca teg ory of the ca tegory o f inv er se semigroups a nd their ho momorphisms and so the meet co mpletion o f K ( S ) is K ( K ( S )) and not just K ( S ). A t this p oint, we want to highlight a c la ss of tra nsitive actions that will play a n impo rtant role b oth here a nd in Se c tio n 4. Remark 3.1 . Let T be an inv erse semigr o up and let e b e any idemp otent in T . W e denote by L e the L -class con taining e . The set L e therefore consists of all elements t ∈ T such that d ( t ) = e . Define a partia l function from T × L e to L e by ∃ a · x iff d ( ax ) = e . This defines a tra nsitive action of T on L e called the (left) Sch¨ utzenb er ger action determine d by the idemp otent e . This is the tra ns itive a ction determined b y the clo sed in verse subsemigro up e ↑ . The structur e of K ( S ) is inextricably linked to the structure of transitive ac tio ns of S . The fo llowing was firs t stated in [1 0]. Prop ositio n 3.2. L et S b e an inverse semigr oup. Every tr ansitive r epr esentation of S is t he re striction of a Sch¨ utzenb er ger r epr esentation of K ( S ) . Pr o of. Let H be a c losed inv erse subse mig roup of S . In the in v erse semigro up K ( S ), the L -class L H of the idemp otent H co ns ists of all A ∈ K ( S ) such that A − 1 ⊗ A = H . Let a ∈ A . Then A = ( aH ) ↑ . It follows that L H consists of pre- cisely the left cos ets of H in S . Let A ∈ L H and co ns ider the pr o duct s ↑ ⊗ A . Then this ag ain b elongs to L H precisely when ( sa ) − 1 sa ∈ H a nd is equal to ( saH ) ↑ . It follows that via the map ι the inv erse semigr oup ac ts on L H precisely as it acts on S/H .  If H and K a re tw o idemp otents of K ( S ) then they are D -related iff ther e exists A ∈ K ( S ) s uch that A − 1 ⊗ A = H and A ⊗ A − 1 = K iff H and K a re conjuga te. Thu s the D - classes of K ( S ) are in bijective co r resp ondence with the conjugacy classes of closed inv erse subsemig r oups. W e ma y , in some sens e, ‘globalize’ the connec tio n b e t ween K ( S ) a nd tra nsitive actions of S . Denote by O ( S ) the catego ry whose ob jects are the right S -spaces H/ S and whose arr ows a re the (right) morphisms. W e now reca ll the following construction [12]. Let S b e an inv er se se mig roup. W e can co nstruct from S a r ight cancellative ca tegory , denoted R ( S ), whose element s are pair s ( s, e ) ∈ S × E ( S ) such that d ( s ) ≤ e . W e r e g ard ( s, e ) a s an a rrow fr o m e to r ( s ) and define a pro duct by ( s, e )( t, f ) = ( st, e ). The following genera lizes Ex ample 2.2.3 of [12]. Prop ositio n 3.3. The c ate gory O ( S ) is isomorphic to the c ate gory R ( K ( S )) . Pr o of. W e obser ve first that a morphism with a transitive space a s its domain is determined by its v alue on any elemen t of that domain. Let φ : U / S → V /S b e a mo rphism. Then φ is determined b y the v a lue taken b y φ ( U ) = ( V a ) ↑ . Now the s ta bilizer S U of U is U itself and the stabilizer S ( V a ) ↑ is ( a − 1 V a ) ↑ . Thu s by Theorem 2 .9 , w e hav e that U ⊆ ( a − 1 V a ) ↑ . Co n versely , if we are given that U ⊆ ( a − 1 V a ) ↑ then we can define a morphism fro m U /S to V / S by U 7→ ( V a ) ↑ . 12 M. V. LA WSON, S. W. M AR GOLIS, AND B. STEI NBER G There is ther efore a bijection betw een mor phisms from U /S to V / S and inclu- sions U ⊆ ( a − 1 V a ) ↑ . W e shall enco de the morphism φ b y the triple ( V , ( V a ) ↑ , U ). Let ψ : V /S → W/S be a mor phism enco ded by the triple ( W, ( W b ) ↑ , V ). The triple enco ding ψ φ is of the form ( W , ( W c ) ↑ , U ) wher e ψ φ ( U ) = ( W c ) ↑ . Thus ( W , ( W b ) ↑ , V )( V , ( V a ) ↑ , U ) = ( W, ( W ba ) ↑ , U ). The pro duct ( W b ) ↑ ⊗ ( V a ) ↑ in K ( S ) is prec isely ( W ba ) ↑ . W e now recall tha t the natural partial o rder in K ( S ) is reverse inclusion. It follows that the tr iple ( V , ( V a ) ↑ , U ) can b e identified with the pair (( V a ) ↑ , U ) where d (( V a ) ↑ ) ≤ U . W e regar d (( V a ) ↑ , U ) as an arrow with domain U and co doma in V . The r esult now follows.  3.2. The inv erse semi group of filters L ( S ) . W e shall now describ e an inv erse subsemigroup of K ( S ). A subset A ⊆ S of an inv erse semigro up S is s aid to be (down) dir e cte d if it is non-empt y and, for each a , b ∈ A , ther e exists c ∈ A such that c ≤ a, b . Clos ed directed sets in a po set are called filters . When this definition is applied to semilattices then we recov er the definition g iven earlier . Lemma 3. 4. The close d dir e cte d subsets ar e pr e cisely the dir e cte d c osets. Pr o of. A directed co set is certainly a closed directed subs et. Let A b e a closed directed subset. W e prove that it is an atlas. Clearly A ⊆ AA − 1 A . Th us we need only check tha t AA − 1 A ⊆ A . Let a, b, c ∈ A . Then since A is direc ted there is d ∈ A such that d ≤ a, b , c . Thus d = dd − 1 d ≤ ab − 1 c a nd so ab − 1 c ∈ A since A is also closed.  Lemma 3.5 . A close d inverse subsemigr oup T of an inverse semigr oup S is dir e cte d if and only if ther e is a filter F ⊆ E ( S ) such that T = F ↑ . Pr o of. Suppose that T = F ↑ . Let a, b ∈ T . Then e ≤ a and f ≤ b fo r so me e, f ∈ F . B ut F is a filter in the semilattice of idempotents and so closed under m ultiplication. Th us ef ∈ F . But then ef ≤ a, b and so T is directed. Let T b e a clos ed dire cted inv e r se subsemig r oup. Put F = E ( S ). L e t e, f ∈ F . Now T is directed and so there is i ∈ T such that i ≤ e , f . T hus i is an idemp otent. But i ≤ ef ≤ e, f a nd so, since F is clo sed, we hav e that ef ∈ F . It fo llows that F is a filter in E ( S ). Clearly F ↑ ⊆ T . Let t ∈ T . Then t − 1 t ∈ T since T is an inv er se s ubsemigroup. But T is directed so ther e exists j ≤ t, t − 1 t . But then j is an idemp otent and so j ≤ t gives that t ∈ F ↑ . Hence T ⊆ F ↑ . Thus T = F ↑ , as required.  Lemma 3.6. If A and B ar e b oth dir e cte d c osets then ( AB ) ↑ is the smal lest dir e cte d c oset c ont aining AB ; it is also t he sm al lest c oset c ontaining AB . Pr o of. The s et ( AB ) ↑ is closed s o we need o nly show it is directed. Let ab, a ′ b ′ ∈ AB . Then there exists c ≤ a, a ′ where c ∈ A and d ≤ b, b ′ where d ∈ B . It fo llows that cd ∈ AB and cd ≤ ab, a ′ b ′ . Thus the set is directed. Now let X b e any coset containing AB . The n X is closed and so ( AB ) ↑ ⊆ X .  The subset of K ( S ) consis ting o f directed cosets is denoted by L ( S ). Prop ositio n 3.7. L et S b e an inverse semigr oup. (i): L ( S ) is an inverse subsemigr oup of K ( S ) . INVERSE SEMIGROUPS AND GR OUPOIDS 13 (ii): The dir e cte d c osets of S ar e pr e cisely the c osets of the close d di r e cte d inverse subsemigr oups of S . (iii): Each element of K ( S ) is t he me et of a subset of L ( S ) c ontaine d in an H -class of L ( S ) . Pr o of. (i) If A, B ∈ K ( S ) then their pro duct is the intersection of all cosets contain- ing AB . But if A, B ∈ L ( S ) then by Lemma 3.6 this in tersection will also be long to L ( S ). Closure under inv er ses is immediate. Thus L ( S ) is an inv erse subsemigr oup of K ( S ). (ii) If A ∈ K ( S ) then A = ( aH ) ↑ = ( a ) ↑ ⊗ H where H = A − 1 ⊗ A and a ∈ A . Thu s A is directed if a nd o nly if H is directed. (iii) Let A ∈ K ( S ) b e a c oset. Define a rela tion ∼ on the set A by a ∼ b iff there ex ists c ∈ A such that c ≤ a, b . W e show that ∼ is an equiv alence relation on A . Clearly ∼ is reflexive and symmetric. It only r emains to prov e that it is transitive. Let a ∼ b and b ∼ c . Then there e x ists x ≤ a, b and y ≤ b, c where x, y ∈ A . In par ticular, x, y ≤ b . Thus z = xy − 1 y = y x − 1 x is the meet of x and y . Since A is a coset xy − 1 y , y x − 1 x ∈ A . It follows that z ≤ a, c . Denote the blo cks of the par tition induced by ∼ on A b y A i where i ∈ I . Ea ch blo ck is direc ted by construction a nd easily seen to b e closed. It follows that ea ch blo ck is a dir ected coset and so A i ∈ L ( S ). W e hav e therefore pr ov e d that A = V i ∈ I A i . It remains to show that A i H A j . T o do this it is enough to compute A − 1 i ⊗ A i and A i ⊗ A − 1 i and observe that these idempo ten ts do not dep e nd o n the suffix i . W e may write A = ( aH ) ↑ for so me clo sed inverse subsemigroup H of S and element a such that d ( a ) ∈ H . Put F = E ( H ) the semilattice of idemp otents of H . Put K = F ↑ and L = ( aK a − 1 ) ↑ , b oth closed directed inv erse s ubsemigroups of S a nd so elements of L ( S ). W e prove that K = A − 1 i ⊗ A i and L = A i ⊗ A − 1 i . F rom A ≤ A i we hav e that H = A − 1 ⊗ A ≤ A − 1 i ⊗ A i and ( aH a − 1 ) ↑ ≤ A i ⊗ A − 1 i . By construction H ≤ K a nd K is in fact the smallest idemp otent of L ( S ) a bove H . It fo llows that K ≤ A − 1 i ⊗ A i and similar ly L ≤ A i ⊗ A − 1 i . It r emains to show that equality holds in ea ch case which means checking tha t K ⊆ A − 1 i ⊗ A i and L ⊆ A i ⊗ A − 1 i . Let k ∈ K and a i ∈ A i . No w k ∈ K ⊆ H and a i ∈ A i ⊆ A . Thus a i k ∈ A . But a i k ≤ a i . Now if a i k ∈ A j then by closure a i ∈ A j and s o we must hav e that a i k ∈ A i . Thus k a − 1 i a i ∈ A − 1 i ⊗ A i and so by clos ure k ∈ A − 1 i ⊗ A i , as requir ed. Let l ∈ L . Let a i ∈ A i . The n A = ( a i H ) ↑ . Thus L = ( a i K a − 1 i ) ↑ . It follows that a − 1 i l a i ∈ K and so a i a − 1 i l ∈ a i K a − 1 i giving l ∈ A i ⊗ A − 1 i . An alternative wa y of proving this re s ult is to o bserve that K is a clo sed inv e r se subsemigroup o f H and so H can b e written as a disjoint union of some o f the left cosets of K . W e can then use this deco mpo s ition to write A itself as a dis jo in t union of left cosets of K .  W e say that an inv erse se mig roup S is me et c omplete if every non-empty subse t of S has a meet. Meet completions of inv erse semigr oups are dis c ussed a t the end of Section 1.4 of [1 1], [10] and most impor tantly in [16]. The meet c o mpletion of an in verse semigro up S is in fact K ( S ) [16]. The inv e rse semigroup S is sa id to hav e a ll dir e cte d me ets if it has meets of a ll non-empty directed subsets. The r esult be low shows that L ( S ) is the dir e cte d meet completion of S in the sa me way that K ( S ) is the meet co mpletion. 14 M. V. LA WSON, S. W. M AR GOLIS, AND B. STEI NBER G Prop ositio n 3.8. L et S b e an inverse semigr oup. Then L ( S ) is t he dir e cte d me et c ompletion of S . Pr o of. W e hav e the embedding ι : S → L ( S ) and o nce aga in each A ∈ L ( S ) is the join of a ll the s ↑ where s ∈ A . This time the set ov er which we are calculating the meet is directed. Le t A = { A i : i ∈ I } b e a directed subs et of K ( S ). Thus for each pair of cosets A i and A j there is a coset A k such that A k ≤ A i , A j . Put A = S i ∈ I A i . It is cle a rly a closed s ubset. If a, b ∈ A then a ∈ A i and b ∈ A j for some i and j . By assumption A i , A j ⊆ A k for some k . Thus a, b ∈ A k . But A k is a directed subset and so ther e exists c ∈ A k such that c ≤ a, b . It follows that A is a closed and dir e c ted subset and so is a directed coset by Lemma 3.4. It is now immediate that A is the meet of the set A . Let θ : S → T be a ho momor- phism to a n in verse semigro up T whic h has a ll meets o f dir ected subsets. Define ψ : K ( S ) → T by ψ ( A ) = V θ ( A ). Then ψ is a homomorphism and the unique one such that ψ ι = θ .  In [17], Lenz constructs a n in verse s e migroup O ( S ) from an inv erse semigroup S , which is the basis for his ´ etale gro upo id asso cia ted with S . The key result for our pa per is the following. Theorem 3.9. The inverse semigr oup L ( S ) is isomorp hic to L enz’s semigr oup O ( S ) . Pr o of. Let F = F ( S ) de no te the s et of directed s u bsets of S . F or A, B ∈ F define A ≺ B iff for each b ∈ B ther e exists a ∈ A such that a ≤ b . This is a preo r der. The asso ciated equiv alence rela tion is given by A ∼ B iff A ≺ B and B ≺ A . W e now make t wo key observ a tions. (1) A ∼ A ↑ . It is easy to chec k that A ↑ is directed. By definition A ≺ A ↑ , wher eas A ↑ ≺ A is immediate. (2) A ↑ ∼ B ↑ iff A ↑ = B ↑ . The r e is only o ne direction needs proving. Supp os e that A ↑ ∼ B ↑ . Let a ∈ A ↑ . Then B ↑ ≺ A ↑ and so there is b ∈ B such that b ≤ a . But then a ∈ B ↑ . Thus A ↑ ⊆ B ↑ . The reverse inclusion is proved similarly . By (1) and (2), it follows that A ∼ B iff A ↑ = B ↑ . As a s e t, O ( S ) = F ( S ) / ∼ . W e hav e therefor e set up a bijection be tw een O ( S ) and L ( S ). Lemma 3 .6 tells us that the multiplication defined in [17] in O ( S ) ensures that this bijection is an isomorphism.  Denote by U ( S ) the category whose ob jects ar e the right S -spaces H /S wher e H is directed a nd whose a rrows are the (right) mo rphisms. W e hav e the following analogue of Prop os itio n 3.3. Prop ositio n 3.10. The c ate gory U ( S ) is isomorphic to the c ate gory R ( L ( S )) . 3.3. P aterson’s ´ etale group o id. Theorem 3.9 brings us to the b eginning of Sec- tion 4 of Lenz’s pap er [17] where he describ es Paterso n’s ´ etale g roup oid. If T is an inv erse se migroup, then it b ecomes a gro upo id when we define a pa rtial binary op eration · , calle d the r estricte d pr o duct , b y ∃ s · t if and only if d ( s ) = r ( t ) in which case s · t = st . P aterson’s g roup oid is pr e cisely ( L ( S ) , · ) equipp ed with a suitable top olog y . The isomorphism functor defined by Lenz fr om L ( S ) to Pater- son’s g roup oid can be very ea sily descr ibe d in terms of the ideas in tro duced in our pap er. Let A ∈ L ( S ). Define P = ( AA − 1 ) ↑ . Then fo r any a ∈ A we have that A = ( P a ) ↑ . Thus we may rega rd A as a right c oset of the close d, directed in verse subsemigroup P . By the dua l of Lemma 2.5(i), we hav e that ( P a ) ↑ = ( P b ) ↑ , where INVERSE SEMIGROUPS AND GR OUPOIDS 15 aa − 1 , bb − 1 ∈ P , if and only if ab − 1 ∈ P if and only if pa = pb for some p ∈ P , where we use the fact that every element of P is ab ov e an idemp otent. The order ed pair ( P , a ) where r ( a ) ∈ P determines the r ight coset ( P a ) ↑ and another such pair ( P, b ) determines the same rig ht coset if and only if p a = pb for so me p ∈ P . This leads to an equiv alence relation and we denote the equiv ale nc e class co n taining ( P, a ) b y [ P , a ]. The iso morphism functor betw een the Lenz group oid L ( S ) and Paterson’s gro up oid is therefore defined by A 7→ [( AA − 1 ) ↑ , a ] where a ∈ A . W e see that Paterson has to work with equiv a le nc e cla s ses b ecause of the non-uniquene s s of cos et-resprese n tatives, and Lenz has to work with equiv alence classes be c a use he works with gener ating sets of filters r ather than with the filters themselves. In our approach, the use of equiv a lence class e s in b oth ca ses is av oided. Recall fro m Section 2.2 , that a transitive S -space X is universal if the stabi- lizer H o f a p oint of X is F ↑ where F is a filter in E ( S ). In other w ords, by Lemma 3.5 the clo sed inv erse s ubsemigroup H is dir ected. It follows that the uni- versal tr ansitive actions o f S a re determined by the directed filter s that are also inv er se subs e migroups. W e shall now des crib e how the structure of the g roup oid ( L ( S ) , · ) r eflects the prope rties of transitive actions of S . In what fo llows, we can just as easily work in the inv er se semig roup as in the group oid. Prop ositio n 3.11. L et S b e an inverse semigr oup. (1) The c onne cte d c omp onents of the gr oup oid L ( S ) ar e in bije ctive c orr esp on- denc e with t he e quivalenc e classes of universal tra nsitive actions of S . (2) L et H b e an identity in L ( S ) . Then the lo c al gr oup G H at H is isomorphi c to the gr oup E ( H ) /σ . Pr o of. (1) The iden tities of L ( S ) a re the closed directed inv erse subs e mig roups of S . Two such identities belo ng to the same connected co mpo nent if and only if they are c onjugate. The result now follows by P rop osition 2.7. (2) Put F = E ( H ) so that H = F . Let A b e in the lo cal gr oup deter mined by H . Then H = ( A − 1 A ) ↑ = ( AA − 1 ) ↑ . Define θ : G H → E ( H ) /σ b y θ ( A ) = σ ( a ) where a ∈ A . W e show fir st that this map is well-defined. Let f ∈ F and let a ∈ A . Then a − 1 f a ∈ A − 1 E ( A ) A ⊆ H A = A and so a − 1 f a ∈ F and af a − 1 ∈ AE ( A ) A − 1 ⊆ AH = A and so a f a − 1 ∈ F . Thus A ⊆ F . Next s uppo s e that a, b ∈ A . Then there is an element c ∈ A such that c ≤ a, b . Thus σ ( a ) = σ ( b ). It follows that θ is well-defined. W e now show that θ defines a bijection. Supp ose that θ ( A ) = θ ( B ). Then aσ b where a ∈ A and b ∈ B . Thus there ex is ts c ∈ F such that c ≤ a, b . It follows tha t c = ac − 1 c = bc − 1 c and so a − 1 ac − 1 c ≤ a − 1 b . But a − 1 ac − 1 c ∈ F and so A = B . Thu s θ is injective. Let a ∈ F . Then a − 1 a, ∈ F a nd so a − 1 a ∈ F . Th us A = ( aH ) ↑ is a well-defined coset and Then ( A − 1 A ) ↑ = H = ( AA − 1 ) ↑ . It follows that A ∈ G H and θ ( A ) = σ ( a ). Thus θ is surjective. Finally we show that θ defines a homomor phis m. Let A, B ∈ G H and a ∈ A and b ∈ B . By Lemma 3.6, A ⊗ B = ( AB ) ↑ and contains ab . Thus θ ( A ) θ ( B ) = σ ( a ) σ ( b ) = σ ( ab ) = θ ( A ⊗ B ).  W e now have the following theorem. 16 M. V. LA WSON, S. W. M AR GOLIS, AND B. STEI NBER G Theorem 3.12. L et S b e an inverse semigr oup. Then L ( S ) explicitly enc o des un i- versal tr ansitive actions of S via its Sch ¨ ut zenb er ger actions, and implicitly enc o des al l tr ansitive actions via its lo c al gr oups. Pr o of. An idempo tent of L ( S ) is just an inv erse subsemigr o up H of S that is als o a filter. Denote by L H the L - c lass of H in the in verse semigroup L ( S ). The elements of L H are just the left cosets of H in S . The inv erse semig r oup L ( S ) acts on the set L H , a Sch¨ utzenberg er ac tio n, and so to o do es S via the map ι of Pr op osition 3.8. This la tter action is equiv alent to the a ction of S on S/H . W e hav e therefore shown that L ( S ) enco des universal transitive actions of S via its Sch¨ utzen b erger actio ns. By Prop os ition 2.15(1) each transitive action o f S o n a set Y is strong ly cov ered by a universal one X . Let H b e a stabilizer o f this universal action of S on X . Then the strong covering is determined by a str o ng congruence which b y Theore m 2.1 6 is determined b y a subgroup of the H -cla ss in L ( S ) containing the idemp otent H ; in other words, by a subg roup of the lo cal g roup determined by the idempotent H .  Finally , the top o logy on the group oid L ( S ) is defined in ter ms of the em b edding S → L ( S ) as follows. Let s ∈ S . Define U s = { A ∈ L ( S ) : s ∈ A } and for s 1 , . . . , s n ≤ s define U s ; s 1 ,...,s n = U s ∩ U c s 1 ∩ . . . ∩ U c s n . Then the sets U s ; s 1 ,...,s n form a basis for a top olog y . 4. Ma trix represent a tio ns of inverse semigroups W e deduce here results o f the thir d author on the finite dimensional irr educible representations o f inv erse semigr oups [34]. There an appro ach based on gro upo id algebras was used, wher e a s here we use results of J. A. Gr een [2, Chapter 6] and the universal prop erty of L ( S ). 4.1. Green’s theorem and primitive ide mp otents. The following theorem summarizes the con ten ts of [2, C ha pter 6]. Le t A b e a ring. A mo dule is as- sumed to b e a left A -mo dule unless otherwise stated. W e also co ns ider only u nitary A -mo dules, that is, A -mo dules M such that AM = M (where AM means the sub- mo dule ge ne r ated by element s am with a ∈ A and m ∈ M ). If A has a unit, then this is the same as saying that the unit acts as the identit y on M . In particular , a simple A -mo dule is an A - mo dule M such AM 6 = 0 and there are no non-zero prop er submo dules of M . If e is an idemp otent o f A and M is an A -mo dule, then eM is a n eAe -mo dule. The functor M 7→ eM is called r est riction and we so metimes deno te it Res e ( M ). It is well known a nd easy to c heck that eM ∼ = Hom A ( Ae, M ), wher e the latter has a left eAe -a ction induced by the rig ht action of eAe on Ae . F or an eAe -mo dule N , define Ind e ( N ) = Ae ⊗ eAe N . The usual hom-tenso r adjunction implies that Ind e is the left adjoint o f Res e . More- ov er, Res e Ind e is isomorphic to the identit y functor on the categor y eAe -mo dules. Indeed, eae ⊗ n 7→ eaen is a n isomorphis m with inv erse n 7→ e ⊗ n . These iso mor- phisms are natural in N . Theorem 4 . 1 (Gre en) . L et A b e a ring and e ∈ A an idemp oten t . INVERSE SEMIGROUPS AND GR OUPOIDS 17 (1) If N is a simple eAe -mo dule, t hen the induc e d m o dule Ind e ( N ) = Ae ⊗ eAe N has a unique maximal submo dule R ( N ) , which c an b e describ e d as the lar gest submo dule of Ind e ( N ) annihilate d by e . Mor e over, the simple mo dule e N = Ind e ( N ) /R ( N ) satisfies N ∼ = e e N . (2) If M is a simple A -mo dule with eM 6 = 0 , then e M is a simple eAe -mo dule and M ∼ = g eM . Let S be an in verse semigr oup a nd s uppo se that e is a minimum idemp otent of S . Then eS e = G e , the maximal subgroup o f S at e , and is a lso the maxima l group ima ge of S . Moreover, S e = G e = eS a nd the ac tion of S on the left of S e factors through the ma x imal group image ho momorphism. Let k be a commutativ e ring with unit. Then ek S e ∼ = k G e and so Green’s theo rem shows that simple k S - mo dules M with eM 6 = 0 a re in bijection with simple k G e -mo dules v ia induction and restrictio n. Mor e over, since k S e = k G e , we hav e tha t Ind e ( N ) = N w ith the action of S induced by the maximal group image ho momorphism. Thus Ind e ( N ) already is a simple k S -mo dule. Let us consider the analogous situation for primitive idempo ten ts. Let e b e a primitive idemp otent of an inv erse semigr oup with 0 . Observe that in this case e S e = G e ∪ { 0 } since e, 0 ar e the o nly idemp otents of eS e and so if s 6 = 0 , then ss − 1 = e = s − 1 s . Thus if k 0 S is the co nt racted semigr o up alg ebra of S (meaning the quotient o f k S by the ideal of scala r multiples of the zero o f S ), then ek 0 S e ∼ = k G e and so aga in by Gr een’s theorem, w e have a bijection b etw een simple k 0 S -modules M with eM 6 = 0 and k G e -mo dules via induction. W e aim to show now that if N is a simple k G e -mo dule, then Ind e ( N ) is alr e ady a simple k 0 S -module. Let L e be the L -class of e . Then since e is primitive, it follows that L e = S e \ { 0 } and s o k 0 S e = k L e where S acts o n the left of k L e via linearly extending the left Sch¨ utzenberger representation. The gro up G e acts freely on the right of L e with orbits the H -classes contained in L e . Thus k 0 S e = k L e is free as a right ek 0 S e = k G e -mo dule. Let T b e a tra ns versal to the H -classes of L e and let N be a k G e -mo dule. Then a s a k -module, Ind e ( N ) = L t ∈ T t ⊗ k N . A fac t w e sha ll use is tha t any element o f L e is primitive and so if t 1 6 = t 2 ∈ T , then t 1 t − 1 1 6 = t 2 t − 1 2 and hence t 1 t − 1 1 t 2 = 0. Lemma 4.2. I f N is a non-zer o k G e -mo dule, then no non-trivial submo dule of Ind e ( N ) is annihilate d by e . Pr o of. Let M b e a no n-zero submo dule of Ind e ( N ). Notice that M is annihilated by e if and only if it is annihilated by the ideal genera ted by e . So let m = P t ∈ T t ⊗ n t (with only finitely many terms non-zero) b e a non-zero ele men t of M . Then there exists t ∈ T with n t 6 = 0. By the observ ation just b efore the pro o f tt − 1 m = t ⊗ n t 6 = 0 and s o tt − 1 do es not annihilate m . But e = t − 1 t ge nerates the s ame idea l a s tt − 1 and so M is not annihilated by e .  As a corolla ry , we obtain fr om Gr een’s Theo rem 4.1 that if N is a simple k G e - mo dule, then Ind e ( N ) is a simple k 0 S -module. Corollary 4. 3. L et S b e an inverse semigr oup, e ∈ E ( S ) a primitive idemp oten t and k a c ommu tative ring with unit. If N is a simple k G e -mo dule, then Ind e ( N ) is a simple k S -mo dule. 18 M. V. LA WSON, S. W. M AR GOLIS, AND B. STEI NBER G If k is a field, then fro m Ind e ( N ) = L t ∈ T t ⊗ k N , we s ee that Ind e ( N ) is finite dimensional if and only if T is finite and N is finite dimensional. 4.2. The main res ult. Suppo se no w that S is an y in verse semigroup and e ∈ E ( S ). Let I e = S eS \ J e be the ideal of elements strictly J - below e . If N is a k G e -mo dule, then let Ind e ( N ) = k 0 [ S/I e ] e ⊗ kG e N = ( k S/k I e ) e ⊗ kG e N . Equiv alently , if L e is the L -class of e , then k L e is a free r ight k G e -mo dule with basis the set of H -cla sses of L e and also it is a left k S -mo dule by means of the actio n of S on the left of L e by partial bijections via the Sch¨ utzenberger repr esentation. Then Ind e ( N ) = k L e ⊗ kG e N . Suppo s e now that the D -clas s of e con tains only finitely many idemp otents; in this case we say that e has finite index in S . Under the hypothesis that e has finite index it is well k nown that if f ∈ E ( S ) with f < e , then S f S 6 = S eS and so f ∈ I e . Th us e is primitive in S/ I e and so Coro llary 4 .3 shows that Ind e ( N ) is simple for any simple k G e -mo dule in this s etting. W e are now ready to constr uct the finite dimensional irreducible r epresentations of an in verse semigroup over a field. This was first ca rried out b y Munn [21], whereas the construction presented her e fir st app eared in [34] where it was deduced as a sp ecial case of a result on ´ etale gr o upo ids. Our appr o ach here uses the inv e rse semigroup L ( S ). Fix a field k . First we cons tr uct a collection o f simple k S -modules . Prop ositio n 4.4. Le t e ∈ E ( L ( S )) have finite index and let N b e a simple k G e - mo dule. Then Ind e ( N ) is a simple k S -mo dule. Mor e over, Ind e ( N ) is finite dimen- sional if and only if N is. Pr o of. The ab ov e discussion shows that Ind e ( N ) is simple a s a k L ( S )-module so we just ne e d to s how that any S -in v aria nt subspace is L ( S )-in v aria n t. In fact, w e show that each element o f L ( S ) acts the same on Ind e ( B ) as some element of S . It will then follow that any S -inv a riant subspace is L ( S )-in v ariant and s o Ind e ( N ) is a simple k S -mo dule. Let T b e a tr ansversal for the orbits of G e on L e . Then T is finite sinc e these orbits are in bijection with R -class es of D e , which in turn are in bijection with idempo ten ts o f D e . Let A ∈ L ( S ) and wr ite A = V d ∈ D s d with s ∈ S and D a directed set. W e claim tha t if t ⊗ n is a n elementary tensor with t ∈ T , then there exists d t ∈ D de p ending only on t (a nd not n ) such that A ( t ⊗ n ) = s d ( t ⊗ n ) for a ll d ≥ d t . B y [11, Section 1 .4, P rop osition 19], we hav e At = V d ∈ D ( s d t ). Since the D - class o f e has only finitely many idemp otents, it fo llows by Theo rem 3.2.16 of [11] that distinct elements o f D are not c omparable in the natural partial order. Since the set { s d t | d ∈ D } is directed, either s d t  L e fo r a ll sufficiently large elements of D or s d t is a n element ℓ of L e independent of d . In the first ca s e At  L e and in the se c ond case At = ℓ . Thus in the fir st c ase, A ( t ⊗ n ) = 0 = s d ( t ⊗ n ) for d large enough, wher eas in the second case A ( t ⊗ n ) = ℓ ⊗ n = s d ( t ⊗ n ) for all d ∈ D . W e conclude d t exists. Since T is finite, we can find d 0 ∈ D with d 0 ≥ d t for all t ∈ R . Then A and s d 0 agree on all elements of the form t ⊗ n with t ∈ T and n ∈ N . But such elements span Ind e ( N ) and so we conclude that A and s d 0 agree on Ind e ( N ). The final statement follows from the previous discussio n.  Note that application of the restriction functor and the fact that Res e Ind e is isomorphic to the iden tit y shows that Ind e ( N ) ∼ = Ind e ( M ) implies N ∼ = M . Also, INVERSE SEMIGROUPS AND GR OUPOIDS 19 if e , f are tw o finite index idemp otents of L ( S ) and e  J f , then f annihilates Ind e ( N ) for a n y k G e -mo dule a nd hence all elements of f , viewed as a filter, anni- hilate Ind e ( N ). On the other hand, no element of the filter e a nnihilates Ind e ( N ). It follows that if e , f are finite index idempotents that ar e no t D - equiv alent, then the mo dules o f the form Ind e ( N ) and Ind f ( M ) a r e never isomo rphic. Clearly , D - equiv alent idemp otents g ive isomor phic co llections o f simple mo dules. Thus, for each D -c la ss with finitely many idemp otents, we get a distinct set o f simple k S - mo dules (up to isomor phism). The following fact is well known and easy to prove. Prop ositio n 4.5. L et k b e a fi eld and V an n -dimensional k -ve ctor sp ac e. The n any semilattic e in End k ( V ) has size at most 2 n . Pr o of. An y idemp otent matrix is diago nalizable and s o any semilattice o f matri- ces is sim ultaneously diagonaliza ble. But the m ultiplicative monoid of k n has 2 n idempo ten ts.  W e can no w co mplete the description of the finite dimensional irre ducible rep- resentations o f an inv erse semigr oup. In the statement of the theo rem b elow, it is worth reca lling that e = H is a finite index, closed directed subsemigro up of S and G e is the gr oup E ( H ) /σ describ ed in Theor em 2 .16. Theorem 4.6. L et k b e a field and S an inverse semigr oup. Then the finite di- mensional simple k S -mo dules ar e pr e cisely those of the form Ind e ( N ) wher e e is a finite index idemp otent of L ( S ) and N is a finite dimensional simple k G e -mo dule. Pr o of. It rema ins to show that ev ery simple k S -mo dule M is of this form. Let θ : S → End k ( V ) b e the corr esp onding irreducible repr esentation. Then T = θ ( S ) is an inv erse s emigroup with finitely many idemp otents and so trivially directed meet complete. Thus θ extends to a homomo r phism θ : L ( S ) → E nd k ( V ) by the universal prop er ty . T rivia lly θ must b e ir reducible as well. Let f b e a minimal non-zero idemp otent of T = θ ( S ) = θ ( L ( S )). Then θ − 1 ( f ) is directed and so has a minim um element e . Suppo se e ′ D e . Suppose e ′′ < e ′ . W e cla im θ ( e ′′ ) = 0. Indeed, choose A ∈ L ( S ) such that A − 1 A = e and AA − 1 = e ′ . Then A − 1 e ′′ A < A − 1 e ′ A = e and so θ ( A − 1 e ′′ A ) = 0. Th us θ ( e ′′ ) = θ ( AA − 1 e ′′ AA − 1 ) = 0. W e conclude θ is injective on the idemp otents of D e . Otherwise, we can find e 1 , e 2 ∈ D e with θ ( e 1 ) = θ ( e 2 ). Then e 1 e 2 ≤ e 1 , e 2 and θ ( e 1 ) = θ ( e 1 e 2 ) = θ ( e 2 ). Thus e 1 = e 1 e 2 = e 2 by the ab ov e claim. W e conclude that e has finite index since T has finitely many idemp o tent s. By c hoice o f e , it now follows that θ factor s through S/I e and he nc e is a k 0 [ S/I e ]- mo dule. Moreov er, e is primitive in S/I e . (If I e = ∅ , then we interpret k 0 [ S/I e ] as k S and e is the minimu m idemp otent.) Since eM = f M 6 = 0 by choice o f f , it follows b y Green’s theor em that N = eM is a simple ek 0 [ S/I e ] e = k G e -mo dule, necessarily finite dimensional. The identit y map N → eM c orresp onds under the adjunction to a no n- zero homomorphism ψ : Ind e ( N ) → M . But we alr eady know that Ind e ( N ) is simple by Prop ositio n 4.4. Sc hur’s lemma then yields that ψ is a n isomorphism. This completes the pro of.  Appendix After an e a rly version of this pap er existed we discov e red tha t J onathon F unk and Pieter Hofstra indep endent ly a rrived at wha t we call universal actions, a nd 20 M. V. LA WSON, S. W. M AR GOLIS, AND B. STEI NBER G which they call tor sors [1]. They show that these co rresp ond exactly to the p oints of the classifying top os of the inv erse semig r oup. F urther c o nnections b et ween our work and their work will b e explored in an upc oming pap er by F unk, Hofstra a nd the third author. In particular, we c o nnect the filter cons tr uction of Paterson’s group oid with the s ob erification of the inductiv e group oid of the inverse semigro up and the so b e rification of the inv er se semigroup. 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