Filtering Germs: Groupoids Associated to Inverse Semigroups

FILTERING GERMS: GROUPOIDS ASSOCIATED TO INVERSE SEMIGROUPS

BECKY ARMSTRONG, LISA ORLOFF CLARK, ASTRID AN HUEF, MALCOLM JONES, AND YING-FEN LIN

Abstract. We investigate various groupoids associated to an arbitrary inverse semigroup with zero. We show that the groupoid of filters with respect to the natural partial order is isomorphic to the groupoid of germs arising from the standard action of the inverse semigroup on the space of idempotent filters. We also investigate the restriction of this isomorphism to the groupoid of tight filters and to the groupoid of ultrafilters.

1. Introduction An inverse semigroup is a set S endowed with an associative binary operation such that for each a ∈ S, there is a unique a∗ ∈ S, called the inverse of a, satisfying aa∗a = a and a∗aa∗ = a∗. The study of étalegroupoids associated to inverse semigroups was initiated by Renault [Ren80, Remark III.2.4]. We consider two well known groupoid constructions: the filter approach and the germ approach, and we show that the two approaches yield isomorphic groupoids. Every inverse semigroup has a natural partial order, and a filter is a nonempty down-directed up-set with respect to this order. The filter approach to groupoid construction first appeared in [Len08], and was later simplified in [LMS13]. Work in this area is ongoing; see for instance, [Bic21, BC20, Cas20]. Every inverse semigroup acts on the filters of its subsemigroup of idempotents. The groupoid of germs associated to an inverse semigroup encodes this action. Paterson pioneered the germ approach in [Pat99] with the introduction of the universal groupoid of an inverse semigroup. Exel’s treatise [Exe08] gives a construction of a groupoid of germs akin to Paterson’s, and introduces the tight groupoid. The germ approach is also used in the construction of Kumjian and Renault’s Weyl groupoid [Kum86, Ren08]. The filter and germ approaches yield the same groupoids up to isomorphism. While this is mentioned in the literature (see [Len08, Theorem 4.13], [MR10, Corollary 5.7], [LMS13, Section 3.3], [LL13, Section 5.1], [Bic21, Remark 2.43], and [Cas20, Corollary 6.8]), the details are omitted or are deeply embedded in proofs. In our main theorem (Theorem 4.1) we give an explicit isomorphism from the groupoid of proper filters to the groupoid of proper germs; we also give a formula for its inverse in terms of the idempotents of the inverse semigroup. The advantage of our approach is that we can then deduce the corresponding results for the groupoids of tight filters and ultrafilters via restriction. This gives a detailed and unified exposition which has already been a arXiv:2010.16113v3 [math.RA] 29 Jul 2021 useful tool in [ACCC21]. We consider a number of groupoids in this work; our notation for these is as follows: • F is the groupoid of proper filters, • T is the groupoid of tight filters, • U is the groupoid of ultrafilters, •G 0 is the groupoid of proper germs, •G tight is Exel’s tight groupoid, and •G ∞ is the groupoid of ultragerms.

Date: 30th July 2021. 2020 Mathematics Subject Classiﬁcation. 06F05, 18B40, 20M18, 22A22. Key words and phrases. Inverse semigroup, groupoid, germs, ﬁlters, patch topology. This research is supported by the Marsden Fund grant 18-VUW-056 from the Royal Society of New Zealand.

1 2 ARMSTRONG, CLARK, AN HUEF, JONES, AND LIN

These groupoids are related via the diagram F ≥ T ≥ U

π π|T π|U

G0 ≥ Gtight ≥ G∞ where G ≥ H means that H is a subgroupoid of G, and the maps are topological groupoid isomorphisms. Note that ≥ is an order relation. The objective of Section 2 is to show that the unit spaces of F and G0 are homeomorphic. In Section 2.1 we present some background on groupoids and inverse semigroups, and in Section 2.2 we define filters of inverse semigroups. We then introduce the various patch topologies in Sec- tion 2.3, and finally in Section 2.4 we describe our homeomorphism. In Section 3 we define the groupoids themselves, and we note key interactions between the structure of filters and germs (see, for instance, Lemma 3.6). In Section 4 we show that F and G0 are isomorphic as topological groupoids (Theorem 4.1), and that our isomorphism restricts to an isomorphism from U to G∞ (Corollary 4.3).

2. Idempotents, filters, and the patch topology The main goal of this paper is to show that the groupoid of proper filters is isomorphic to the groupoid of proper germs, and that the groupoid of ultrafilters is isomorphic to the groupoid of ultragerms. In this section, we show that the unit spaces of these pairs of groupoids are homeomorphic with respect to the patch topology. 2.1. Preliminaries. A groupoid is a set G with a subset G(2) ⊆ G ×G, a composition (α, β) 7→ αβ from G(2) to G and an inversion γ 7→ γ−1 on G such that, for all α, β, γ ∈ G, (i)( γ−1)−1 = γ and (γ−1, γ) ∈ G(2), (ii) if (α, β), (β, γ) ∈ G(2), then (α, βγ), (αβ, γ) ∈ G(2) and α(βγ) = (αβ)γ, and (iii) if (γ, η) ∈ G(2), then γ−1γη = η and γηη−1 = γ. The units of a groupoid G are the elements of the unit space G(0) := {γ−1γ : γ ∈ G}. The source map d: G → G is defined by d(γ) := γ−1γ, and the range map r: G → G is defined by r(γ) := γγ−1. A subset A of G is called a local bisection if d|A : A → G is injective, or, equivalently, if r|A : A → G is injective. A subset H of G is a subgroupoid if, for all γ ∈ H and (α, β) ∈ (H × H) ∩ G(2), we have γ−1, αβ ∈ H. Given groupoids G and H, a bijection φ: G → H is called a groupoid isomorphism if for all (α, β) ∈ G(2), we have (φ(α), φ(β)) ∈ H(2) and φ(αβ) = φ(α)φ(β). We call a groupoid G topological if G is endowed with a topology, with respect to which composition and inversion are continuous (and consequently, so are d and r). A topological groupoid G is étale if its source map (or, equivalently, its range map) is a local homeomorphism. A basis B for the topology on a topological groupoid G is called étale if, for all O,N ∈ B, we have O−1,ON ∈ B and O−1O ⊆ G(0). The topology on G has an étalebasis if and only if the groupoid G is étale[BS19, Proposition 6.6]. If G and H are topological groupoids and φ: G → H is a homeomorphism and a groupoid isomorphism, then we call φ a topological groupoid isomorphism. Assumption. Throughout, let S be an inverse semigroup containing an element 0 satisfying 0a = a0 = 0, for all a ∈ S. Note that even if S doesn’t contain such an element, a 0 can always be adjoined.1 The multiplication and inversion operations in S satisfy (a∗)∗ = a and (ab)∗ = b∗a∗, for all a, b ∈ S. The natural partial order on S is the relation a ≤ b ⇐⇒ a = aa∗b, which satisfies • 0 ≤ a, • a ≤ b ⇐⇒ a∗ ≤ b∗, • a ≤ b and c ≤ d =⇒ ac ≤ bd, and • (ab)∗ab ≤ b∗b,

1Quoting [Exe09]: “. . . one may wonder why in the world would anyone want to insert a zero in an otherwise well behaved semigroup. Rather than shy away from inverse semigroups with zero, we will assume that all of them contain a zero element, not least because we want to keep a close eye on this exceptional element.” FILTERING GERMS: GROUPOIDS ASSOCIATED TO INVERSE SEMIGROUPS 3 for all a, b, c, d ∈ S. An element e ∈ S is called an idempotent if ee = e. The set E of idempotents in S satisﬁes E = {s∗s : s ∈ S}. For all e, f ∈ E, the greatest lower bound of {e, f} is ef. The set E is a commutative inverse semigroup. For all e ∈ E and a ∈ S, we have • 0 ∈ E, • e∗ = e, • a∗ea ∈ E, • ae, ea ≤ a, and • a ≤ e =⇒ a ∈ E. We refer to [Pet84, Law98] for extensive treatments of inverse semigroups.

2.2. Filters, idempotent filters, and filters of idempotents. Given A ⊆ S, we write A↑ := {b ∈ S : a ≤ b for some a ∈ A} and A↓ := {b ∈ S : b ≤ a for some a ∈ A}. For each a ∈ S, we write a↑ := {a}↑ and a↓ := {a}↓. For all e ∈ E, we have e↓ ⊆ E. We say that A is down-directed if, for all a, b ∈ A, there is c ∈ A such that c ≤ a, b, and we say that A is an up-set if A↑ = A.A filter in S is a nonempty down-directed up-set. A filter F is proper if 0 ∈/ F . A filter U that is maximal among proper filters is called an ultrafilter. Ultrafilters are prevalent in S by a standard appeal to Zorn’s lemma. The sets of filters, proper filters, and ultrafilters in S are denoted by L, F, and U, respectively. A filter F in S is called idempotent if E ∩ F 6= ∅. For any subset H of L, we denote by H(0) the set of filters in H that are idempotent. Since E is an inverse semigroup containing 0, it has its own set F(E) of proper filters and set U(E) of ultrafilters, which we call E-filters and E-ultrafilters, respectively.

2.3. Patch topology. Given a set Y , we write A ⊆fin Y when A is a finite subset of Y . For all ↓ s ∈ S and all T ⊆fin s , we define

Fs := {F ∈ F : s ∈ F } and Fs:T := {F ∈ F : s ∈ F ⊆ S \ T }.

Then (F ) ↓ is a basis for a topology [Len08, Proposition 4.1], called the patch topology s:T s∈S,T ⊆ﬁns (0) (0) on F. The collection (F ∩ F ) ↓ is a basis for the patch topology on F , but it will s:T s∈S,T ⊆ﬁns be helpful in computations to have a basis indexed only by idempotents.

(0) Lemma 2.1. The collection (F ) ↓ is a basis for the patch topology on F . e:X e∈E,X⊆fine ↓ (0) Proof. Fix s ∈ S and let T ⊆fin s . Suppose that F ∈ F ∩ Fs:T . It suffices to find e ∈ E and ↓ (0) (0) ↓ X ⊆fin e such that F ∈ Fe:X ⊆ F ∩ Fs:T . Since F ∈ F , there exists e ∈ E ∩ F . Since e ⊆ E, we may assume without loss of generality that e ≤ s. Let X := {et∗t : t ∈ T }. We show that F ∈ Fe:X . Since e ∈ F , it remains to show that F ⊆ S \ X. Suppose, looking for a contradiction, that there is t ∈ T such that et∗t ∈ F . We know that e = es because e ≤ s. Also, st∗t = t because t ∈ s↓. Hence et∗t = (es)t∗t = et ≤ t, and so t ∈ F . However, F ⊆ S \ T , so we have a contradiction. Therefore, F ⊆ S \ X, and so F ∈ Fe:X . (0) Now we show that Fe:X ⊆ F ∩ Fs:T . Fix G ∈ Fe:X . Since e ≤ s, we know s ∈ G, and G ∈ F(0) because e is an idempotent. It remains to show that G ⊆ S \ T . Suppose, looking for a contradiction, that t ∈ G for some t ∈ T . Since G contains an idempotent, we know that G is closed under inversion and multiplication (see, for example, [Law10a, Lemma 2.9]). Thus e, t ∈ G implies that et∗t ∈ G. However, et∗t ∈ X and G ⊆ S \ X, which is a contradiction. Hence, (0) G ⊆ S \ T , as required. Therefore, F ∈ Fe:X ⊆ F ∩ Fs:T . ↓ Analogously, for all e ∈ E and all X ⊆fin e , we define

Fe := {ξ ∈ F(E): e ∈ ξ} and Fe:X := {ξ ∈ F(E): e ∈ ξ ⊆ E \ X}.

The collection (F ) ↓ is a basis for a topology on F(E)[Exe08, Law12, EP16], called e:X e∈E,X⊆ﬁne the patch topology on F(E). The closure of U(E) in F(E) with respect to the patch topology is denoted by T(E), and this coincides with the set T(E) mentioned in the proof of [Law12, Proposition 2.25]. 4 ARMSTRONG, CLARK, AN HUEF, JONES, AND LIN

(0) 2.4. Hausdorff unit space. In Section 3 we define the groupoids F and G0 whose unit spaces F (0) (0) and G0 can both be identified with F(E). First we identify F with F(E) using a generalisation of the map given in [Law12, Lemma 2.18]. Proposition 2.2. There is a bijection : F(0) → F(E) given by (F ) = F ∩ E, with inverse given by −1(ξ) = ξ↑. Moreover, is a homeomorphism of Hausdorff spaces with respect to the patch topology, and the restriction |U(0) is a homeomorphism onto U(E). Proof. To see that is well defined, note that for each F ∈ F(0), we have F ∩ E ∈ F(E), because e↓ ⊆ E for each e ∈ E. Note also that for each ξ ∈ F(E), we have ξ↑ ∈ F(0). To see that is bijective with inverse ξ 7→ ξ↑, it suffices to show that F = (F )↑ and ξ = (ξ↑), for all F ∈ F(0) and ξ ∈ F(E). Fix F ∈ F(0). Toward F ⊆ (F )↑, we take x ∈ F . Since F ∈ F(0), there is some e ∈ (F ). Since F is down-directed, there exists f ∈ F such that f ≤ e, x. Now f ∈ E↓ = E, so f ∈ (F ), and thus x ∈ (F )↑. For the reverse containment, let x ∈ (F )↑, and choose e ∈ F ∩ E such that e ≤ x. It follows that x ∈ F , and thus F = (F )↑. Now fix ξ ∈ F(E). Since ξ ∈ F(E) is an up-set in E, we have (ξ↑) = ξ↑ ∩ E = ξ ∩ E = ξ, as required. ↓ To see that is a homeomorphism, observe that (Fe:X ) = Fe:X , for all e ∈ E and all X ⊆fin e , so Lemma 2.1 implies the result. Note that F(E) is Hausdorff with respect to the patch topology by [Law12, Lemma 2.22] (which applies because E is a meet-semilattice). Since is a homeomorphism, F(0) is Hausdorff as well. (0) (0) To see that |U(0) is a bijection from U to U(E), it suffices to show that (U ) = U(E), because : F(0) → F(E) is a bijection. Fix U ∈ U(0). Suppose that (U) ⊆ ξ ∈ F(E). Then U ⊆ −1(ξ), but U is an ultrafilter, so U = −1(ξ). Hence (U) = ξ, and so (U) ∈ U(E). A (0) similar argument yields the reverse inclusion, so (U ) = U(E).

3. Groupoids associated to inverse semigroups Given a map x 7→ x∗ on a set X, we define A∗ := {a∗ : a ∈ A}, for all A ⊆ X. Given X(2) ⊆ X×X and a map (x, y) 7→ xy from X(2) to X, we define AB := {ab :(a, b) ∈ (A×B)∩X(2)}, xB := {x}B, and Ay := A{y}, for all A, B ⊆ X and x, y ∈ X. 3.1. Groupoids of filters. For all F,G ∈ L, we define F · G := (FG)↑ ⊆ S. Then L is an inverse semigroup with multiplication given by (F,G) 7→ F · G and inversion given by F 7→ F ∗. By [LMS13, Theorem 3.8], L is isomorphic to Lenz’s semigroup O(S) defined in [Len08, Theorem 3.1]. By restricting multiplication in L to L(2) := {(F,G) ∈ L × L : F ∗ · F = G · G∗}, we obtain a groupoid (see [Law98, Proposition 4 of Section 3.1]). Under these operations, the set F forms a subgroupoid of L, which we call the groupoid of proper filters. (This is the groupoid denoted by L(S) in [LL13].) It is a consequence of [Law98, Proposition 1 of Section 9.2] that U is a subgroupoid of F, which we call the groupoid of ultrafilters. The groupoid of ultrafilters is the Weyl groupoid from [Bic21, Definition 2.42], which is denoted by G(S) in [Law12, LL13]. The unit space of F is precisely the set F(0) of idempotent proper filters, and the unit space of U is U(0). The following lemma is used to prove Lemma 3.2. A more general version can be found in [Law93, Proposition 1.4(ii)] (see also [Law10a, Lemmas 2.8 and 2.11]). Lemma 3.1. Fix F,G ∈ F. (a) For all s ∈ F , we have F = (sd(F ))↑. (b) If F ∩ G 6= ∅ and d(F ) = d(G), then F = G. Proof. For part (a), fix s, x ∈ F . Then there exists y ∈ F such that y ≤ x, s. Thus sy∗y = y ≤ x, and so x ∈ (sd(F ))↑. Since sd(F ) ⊆ F and F is an up-set, the reverse containment is clear. For part (b), suppose that s ∈ F ∩ G and d(F ) = d(G). Then part (a) gives ↑ ↑ F = (sd(F )) = (sd(G)) = G. The groupoid F of proper filters has Hausdorff unit space F(0) with respect to the patch topology, by Proposition 2.2. We will show that F is an étalegroupoid with respect to the patch topology, FILTERING GERMS: GROUPOIDS ASSOCIATED TO INVERSE SEMIGROUPS 5 but to see this, we will first consider a coarser topology on F generated by the subcollection (Fs)s∈S, which is a basis by [Bic21, Cas20]. Interestingly, in the subspace U of ultrafilters, the topology generated by the basis (Fs ∩ U)s∈S is the entire patch topology (see Proposition 4.4). Lemma 3.2. For all s, t ∈ S, we have ∗ (a) Fs = Fs∗ , (b) FsFt = Fst, (c) Fs is a local bisection in F, (0) (d) d(Fs) = Fs∗s ⊆ F , and (e)( Fs)s∈S is an étalebasis for the topology on the groupoid F. Proof. Fix s, t ∈ S. Part (a) follows from the definitions and that (s∗)∗ = s. Our argument for part (b) is inspired by the proof of [Law12, Lemma 2.10(4)]. It is clear that FsFt ⊆ Fst. Fix ∗ ↑ ↑ ↑ H ∈ Fst. Put F := (s(td(H)t ) ) and G := (td(H)) . The subsets F and G are filters because they are upward closures of down-directed sets. Since s∗s ∈ E, we have t∗t ≥ t∗(s∗s)t = (st)∗st ∈ d(H), ∗ ∗ ∗ and so t t ∈ d(H). Thus, t = tt t ∈ G, so G ∈ Ft. Since tt ∈ E, we have s∗s ≥ s∗stt∗ ≥ tt∗s∗stt∗ = t(st)∗stt∗ ∈ td(H)t∗, ∗ ∗ ↑ ∗ and so s s ∈ (td(H)t ) . Thus, s = ss s ∈ F , so F ∈ Fs. We now show that d(F ) ⊆ r(G). Fix x ∈ F . We must show that x∗x ∈ r(G). Choose h ∈ H such that sth∗ht∗ ≤ x. Since h, st ∈ H, we may assume without loss of generality that h ≤ st, and so x ≥ sth∗ht∗ ≥ ht∗. Thus, x∗x ≥ (ht∗)∗ht∗ = th∗ht∗ = th∗hh∗ht∗ = th∗h(th∗h)∗ ∈ r(G), and so x∗x ∈ r(G). Now we show that r(G) ⊆ d(F ). Fix y ∈ G. We must show that yy∗ ∈ d(F ). Choose h ∈ H such that y ≥ th∗h. Since sth∗ht∗ ∈ F and th∗ht, s∗s ∈ E, we have yy∗ ≥ th∗h(th∗h)∗ = th∗ht∗ ≥ th∗ht∗s∗sth∗ht∗ = (sth∗ht∗)∗sth∗ht∗ ∈ d(F ), and so yy∗ ∈ d(F ). Therefore, d(F ) = r(G). To see that F · G = H, fix x ∈ F · G. Then there exist h1, h2 ∈ H such that ∗ ∗ ∗ x ≥ (sth1h1t )(th2h2). ∗ ∗ Since h1, h2, st ∈ H, there exists h ∈ H such that h ≤ h1, h2, st. Now, since h1h1, t t ∈ E, we have ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ x ≥ sth1h1t th2h2 = stt th1h1h2h2 = sth1h1h2h2 ≥ sth h ≥ hh h = h ∈ H, and hence x ∈ H. For the reverse inclusion, fix h ∈ H. We may assume that h ≤ st, so a similar computation yields h = (sth∗ht∗)(th∗h) ∈ F · G. Hence, F · G = H. It remains to show that F and G are proper filters, so that H = F · G ∈ FsFt. If either F or G contains 0, then so must H, because H = F · G. But H is proper, so neither F nor G contains 0, completing the proof. ∗ (0) Part (c) follows from Lemma 3.1(b). For part (d), note that s s ∈ E, so Fs∗s ⊆ F . The inclusion d(Fs) ⊆ Fs∗s follows from parts (a) and (b). Part (b) implies that for the reverse (2) inclusion, it is enough to show that Fs∗ Fs ⊆ d(Fs). Fix (F,G) ∈ (Fs∗ × Fs) ∩ F , so that ∗ ∗ ∗ F · G ∈ Fs∗ Fs. Then d(F ) = d(G ) and F,G ∈ Fs∗ , so F = G by Lemma 3.1(b). Thus, ∗ F · G = G · G = d(G) ∈ d(Fs). Part (e) is a consequence of parts (a), (b), and (d). By [BS19, Proposition 6.6], a groupoid endowed with a topology is étaleif and only if the topology has an étalebasis. Thus, Proposition 3.3 is immediate from Lemma 3.2(e). Proposition 3.3. The groupoid F of proper filters is étalewith respect to the topology generated by (Fs)s∈S.

Remark 3.4. Although the topology generated by (Fs)s∈S is nice enough to make F étale,it can be shown that F(0) is non-Hausdorff, provided there are distinct e, f ∈ E such that ef 6= 0. If the intention is to find a groupoid with a Hausdorff unit space, Proposition 4.4 tells us that it suffices to restrict to the groupoid U of ultrafilters. We would rather keep all proper filters on hand for 6 ARMSTRONG, CLARK, AN HUEF, JONES, AND LIN now, so instead of taking a smaller groupoid we refine the topology to the patch topology on the same groupoid. Corollary 3.5. The groupoid F of proper filters has a Hausdorff unit space and is étalewith respect to the patch topology. Proof. The groupoid F of proper filters is a topological groupoid with respect to the patch topology by [Len08, Proposition 4.3]. Since the patch topology is finer than the topology generated by (0) (Fs)s∈S, F is étaleby Proposition 3.3, and F is Hausdorff by Proposition 2.2. Henceforth, the groupoid F of proper filters is endowed with the patch topology. 3.2. Groupoids of germs. The initial data used to construct a groupoid of germs employs the notion of an ‘action’ of S on a nice topological space X [EP16, Definition 3.1]. We are exclusively interested in the ‘standard’ action β of S on F(E) described in [EP16, Section 3], so we neglect the details of inverse semigroup actions for brevity, introducing structure only as necessary. Define Ω := {(s, ξ) ∈ S × F(E): ξ ∈ Fs∗s}, and define a relation ∼ on Ω by (s, ξ) ∼ (t, η) if and only if there is some e ∈ ξ = η such that se = te. The relation ∼ is an equivalence relation on Ω. We denote the equivalence class of (s, ξ) by [s, ξ]. Define G0 := Ω/ ∼. For each s ∈ S and ξ ∈ Fs∗s, define ∗ βs(ξ) := {f ∈ E : ses ≤ f for some e ∈ ξ}, as per [EP16, Equation (3.4)]. Notice that βs(ξ) ∈ Fss∗ . Define (2) G0 := {([s, ξ], [t, η]) ∈ G0 × G0 : ξ = βt(η)} ⊆ G0 × G0. (2) The set G0 with the composable pairs G0 and the operations −1 ∗ ([s, βt(η)], [t, η]) 7→ [s, βt(η)][t, η] := [st, η] and [s, ξ] 7→ [s, ξ] := [s , βs(ξ)] forms a groupoid, which we call the groupoid of proper germs. The unit space of G0 is the set (0) G0 = {[e, ξ]: ξ ∈ Fe}, which may be identified with F(E) via the bijection [e, ξ] 7→ ξ from [EP16, Equation (3.9)]. Due to [Exe08, Proposition 12.8] and [EP16, Proposition 3.5], we have that

βs(Fs∗s ∩ T(E)) ⊆ T(E) and βs(Fs∗s ∩ U(E)) ⊆ U(E), for all s ∈ S. (0) Thus, T(E) and U(E) are invariant subsets of G0 , and the corresponding reductions Gtight and G∞ are subgroupoids of G0 with unit spaces T(E) and U(E), respectively. Note that Gtight is Exel’s tight groupoid from [Exe08, Theorem 13.3]. We call G∞ the groupoid of ultragerms. For each s ∈ S and each open subset A ⊆ Fs∗s, define Θ0(s, A) := {[s, ξ] ∈ G0 : ξ ∈ A}. The collection of all such sets Θ0(s, A) is a basis for a topology on G0, with respect to which G0 is a locally compact étale groupoid with a Hausdorff unit space [EP16, Section 3].

Lemma 3.6. Fix F ∈ F, and define ξF := d(F ) ∩ E. (a) We have ξF ∈ F(E) and ξU ∈ U(E), for all U ∈ U. (b) For all s, t ∈ F , we have [s, ξF ] = [t, ξF ]. (c) For all s ∈ F , we have βs(ξF ) = ξF ∗ . Proof. For part (a), fix F ∈ F and U ∈ U. Then d(F ) ∈ F(0) and d(U) ∈ U(0), so Proposition 2.2 implies that ξF = (d(F )) ∈ F(E) and ξU = (d(U)) ∈ U(E). ∗ For part (b), fix s, t ∈ F . Choose u ∈ F such that u ≤ s, t. Observe that u u ∈ ξF , so it suffices to show that su∗u = tu∗u, which holds since su∗u = u = tu∗u by the definition of ≤. ∗ For part (c), fix s ∈ F . Since ξF ∗ = d(F ) ∩ E = r(F ) ∩ E, it suffices to show that βs(ξF ) = ∗ ∗ ∗ r(F ) ∩ E. Fix f ∈ βs(ξF ) ⊆ E. Then ses ≤ f for some e ∈ ξF . Since s ∈ F and e ∈ ξF = d(F ) ∩ E, we have f ≥ ses∗ ∈ F · d(F ) · F ∗ = F · F ∗ = r(F ), and hence f ∈ r(F ) ∩ E. It remains to show that r(F ) ∩ E ⊆ βs(ξF ). Fix x ∈ r(F ) ∩ E. Since r(F ) = (FF ∗)↑, there exists a ∈ F such that aa∗ ≤ x. Choose b ∈ F such that b ≤ a, s. Observe ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ that s aa s ≥ b bb b = b b ∈ d(F ), and so s aa s = (a s) a s ∈ d(F ) ∩ E = ξF . Now notice that s(s∗aa∗s)s∗ = ss∗(aa∗)(ss∗) = ss∗ss∗aa∗ = ss∗aa∗ ≤ aa∗ ≤ x, FILTERING GERMS: GROUPOIDS ASSOCIATED TO INVERSE SEMIGROUPS 7 from which it follows that x ∈ βs(ξF ), by the definition of βs.

4. Mapping filters to germs

Recall from Lemma 3.6 that for each F ∈ F, we have ξF := d(F )∩E ∈ F(E) and [s, ξF ] = [t, ξF ] for all s, t ∈ F .

Theorem 4.1. The map π : F → G0 given by π(F ) := [s, ξF ] for any s ∈ F is a topological groupoid isomorphism, with inverse given by π−1([s, ξ]) = (sξ)↑. Proof. We begin by showing that π is injective. Suppose that π(F ) = π(G) for some F,G ∈ F. Choose any s ∈ F and t ∈ G. Then [s, ξF ] = π(F ) = π(G) = [t, ξG], so there exists e ∈ ξF = ξG such that se = te. Since e ∈ ξF ⊆ d(F ), we have se ∈ F ·d(F ) = F ; similarly, since e ∈ ξG ⊆ d(G), we have se = te ∈ G · d(G) = G. Thus, F,G ∈ Fse. Recall from Proposition 2.2 that the map : H 7→ H ∩ E is a bijection from F(0) to F(E), with inverse given by −1(ξ) = ξ↑. Thus

ξF = ξG =⇒ (d(F )) = (d(G)) =⇒ d(F ) = d(G).

Since Fse is a local bisection by Lemma 3.2(c), we deduce that F = G. −1 Next we show that π is surjective. Fix [s, ξ] ∈ G0. Since ξ ∈ Fs∗s, we have (ξ) ∈ Fs∗s = d(Fs) −1 by Proposition 2.2 and Lemma 3.2(d). Take F ∈ Fs such that d(F ) = (ξ). Then s ∈ F and −1 ξF = (d(F )) = ( (ξ)) = ξ. It follows that π(F ) = [s, ξ], and hence π is a bijection. Moreover, Lemma 3.1(a) gives π−1([s, ξ]) = π−1(π(F )) = F = (sd(F ))↑ = (s−1(ξ))↑ = (sξ↑)↑ = (sξ)↑. Now we show that π is a groupoid isomorphism. Fix (F,G) ∈ F(2). We must show that (2) (π(F ), π(G)) ∈ G0 and π(F · G) = π(F )π(G). Let s ∈ F and t ∈ G. Then π(F ) = [s, ξF ] and (2) (2) π(G) = [t, ξG]. By the deﬁnition of G0 , we need ξF = βt(ξG). Since (F,G) ∈ F , we have d(G∗) = r(G) = d(F ), and hence Lemma 3.6(c) implies that ∗ βt(ξG) = ξG∗ = (d(G )) = (d(F )) = ξF .

Now, since st ∈ F · G and ξF ·G = d(F · G) ∩ E = d(G) ∩ E = ξG, we have

π(F · G) = [st, ξF ·G] = [st, ξG] = [s, ξF ][t, ξG] = π(F )π(G).

To see that π is continuous, fix s ∈ S and take any open set A ⊆ Fs∗s. Then Θ0(s, A) is an arbitrary basic open set in G0. A routine argument shows that −1 −1 −1 π (Θ0(s, A)) = Fs ∩ d ( (A)). Since d is continuous and Proposition 2.2 implies that −1(A) is open in F(0), we deduce that −1 −1 −1 −1 −1 d ( (A)) is open in F. Since Fs is also open, π (Θ0(s, A)) = Fs ∩ d ( (A)) is open in F. ↓ Finally, to see that π : F → G0 is open, fix s ∈ S and let T ⊆fin s , so that Fs:T is a basic open set in F. Since both d and are open maps by Corollary 3.5 and Proposition 2.2 respectively, we deduce that π(Fs:T ) = Θ0(s, (d(Fs:T ))) is a basic open set in G0, and hence π is open.

The map π : F → G0 restricts nicely to the ultraﬁlters.

Lemma 4.2. We have π(U) = G∞.

Proof. Fix U ∈ U and let s ∈ U. Then π(U) = [s, ξU ]. By Lemma 3.6(a), we have ξU ∈ U(E), and hence π(U) ∈ G∞. For the reverse containment, ﬁx [s, ξ] ∈ G∞ ⊆ G0. Since π maps F onto G0, there is some U ∈ F such that π(U) = [s, ξ]. Fix t ∈ U. Then [t, ξU ] = π(U) = [s, ξ]. In particular, ξ = ξU , and hence d(U) ∩ E = ξU = ξ ∈ U(E). Therefore, Proposition 2.2 implies that d(U) ∈ U(0), and so U = U · d(U) ∈ U, because U is an ideal in F by [Bic21, Proposition 2.41]. Thus, [s, ξ] = π(U) ∈ π(U). Corollary 4.3 is immediate from Lemma 4.2 and Theorem 4.1, since the patch topologies on U and G∞ are the subspace topologies relative to the patch topologies on F and G0, respectively.

Corollary 4.3. The restriction π|U : U → G∞ is a topological groupoid isomorphism with respect to the patch topologies. 8 ARMSTRONG, CLARK, AN HUEF, JONES, AND LIN

For all s ∈ S, deﬁne Us := Fs ∩ U. We generalise [LL13, Proposition 5.18].

Proposition 4.4. The collection (Us)s∈S is a basis for the patch topology on U.

Proof. By Corollary 4.3, it suﬃces to show that (π(Us))s∈S is a basis for the topology on G∞. For each s ∈ S and open set A ⊆ Us∗s, deﬁne Θ∞(s, A) := {[s, ξ]: ξ ∈ A}. The collection of all such sets Θ∞(s, A) is a basis for the topology on G∞. Observe that

π(Us) = {[s, ξU ]: U ∈ Us} = {[s, ξ]: ξ ∈ Us∗s} = Θ∞(s, Us∗s), and so (π(Us))s∈S = (Θ∞(s, Us∗s))s∈S. Thus it suffices to show that (Θ∞(s, Us∗s))s∈S is a basis for the topology on G∞. Fix s ∈ S and take any open set A ⊆ Us∗s. Let [s, ξ] ∈ Θ∞(s, A). Since the collection of sets of the form Θ∞(s, A) is a basis for the topology on G∞, it suffices to find t ∈ S such that [s, ξ] ∈ Θ∞(t, Ut∗t) ⊆ Θ∞(s, A). It follows from [Law12, Lemma 2.26] (which S applies to the collection of idempotents in any inverse semigroup) that A = e∈X Ue for some X ⊆ E. Thus, there is some e ∈ X such that ξ ∈ Ue, and so [s, ξ] ∈ Θ∞(s, Ue) ⊆ Θ∞(s, A). By putting t := se and observing that t∗t = s∗se ∈ ξ and t∗t ≤ e, we see that

[s, ξ] ∈ Θ∞(t, Ut∗t) ⊆ Θ∞(s, Ue) ⊆ Θ∞(s, A).

Remark 4.5. Knowing that (Us)s∈S is a basis for the patch topology on U leads to the following characterisation of convergence of nets in U: for any net (Uk)k∈K ⊆ U and any U ∈ U,(Uk)k∈K converges to U if and only if, for each u ∈ U, there exists ku ∈ K such that u ∈ Uk for all k ku. −1 Remark 4.6. Define T := π (Gtight), where Gtight is Exel’s tight groupoid given in [Exe08, The- orem 13.3]. Since π : F → G0 is a topological groupoid isomorphism and Gtight is a subgroupoid of G0, T is a topological subgroupoid of F, which is topologically isomorphic to Gtight. By [LL13, Lemma 5.9], T is the groupoid denoted by Gt(S) in [LL13]. Furthermore, T is the reduction of L with respect to a certain coverage notion referred to in [Cas20, Corollary 6.8]. Remark 4.7. Let L(E) be the set of filters in the inverse semigroup E, which we can identify with the spectrum of E defined in [Exe08, Definition 10.1]. There is a groupoid G of germs associated to (S, L(E)), as G0 is to (S, F(E)). The groupoid G of germs can be identified with Paterson’s universal groupoid Gu [Ste10, Definition 5.14], and π extends to a topological isomorphism λ from L to G [Cas20, p5]. Moreover, the map s 7→ s↑ from S to L is a faithful homomorphism of inverse semigroups. Therefore, the diagram from Section 1 extends as follows: ↑ S L ≥ F ≥ T ≥ U

λ π π|T π|U ∼ Gu = G ≥ G0 ≥ Gtight ≥ G∞ Remark 4.8. The set E is a semilattice in the sense that ef is the greatest lower bound of {e, f}, for all e, f ∈ E. The semilattice E is compactable in the sense of [Law10b] if and only if T(E) = U(E), by [Law10b, Theorem 2.5]. When T(E) = U(E), Gtight coincides with G∞. Therefore, E is compactable if and only if the map π|U is a topological groupoid isomorphism from the groupoid U of ultraﬁlters to Exel’s tight groupoid Gtight.

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(B. Armstrong) Mathematical Institute, WWU Munster,¨ Einsteinstr. 62, 48149 Munster,¨ GER- MANY Email address: [email protected]

(L.O. Clark, A. an Huef, and M. Jones) School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, NEW ZEALAND Email address: lisa.clark, astrid.anhuef, [email protected]

(Y.-F. Lin) Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast, BT7 1NN, UNITED KINGDOM Email address: [email protected]