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Filtering Germs: Groupoids Associated to Inverse Semigroups

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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 2 S, there is a unique a∗ 2 S, called the inverse of a, satisfying aa∗a = a and a∗aa∗ = a∗: The study of ´etalegroupoids 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 men- tioned 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 isomor- phism 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 ap- proach 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 1 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 πjT πjU G0 ≥ Gtight ≥ G1 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 G1 (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 2 G, (i)( γ−1)−1 = γ and (γ−1; γ) 2 G(2), (ii) if (α; β); (β; γ) 2 G(2), then (α; βγ); (αβ; γ) 2 G(2) and α(βγ) = (αβ)γ, and (iii) if (γ; η) 2 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) := fγ−1γ : γ 2 Gg. 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 djA : A !G is injective, or, equivalently, if rjA : A !G is injective. A subset H of G is a subgroupoid if, for all γ 2 H and (α; β) 2 (H × H) \G(2), we have γ−1; αβ 2 H. Given groupoids G and H, a bijection φ: G!H is called a groupoid isomorphism if for all (α; β) 2 G(2), we have (φ(α); φ(β)) 2 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 ´etale 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 ´etale if, for all O; N 2 B, we have O−1; ON 2 B and O−1O ⊆ G(0). The topology on G has an ´etalebasis if and only if the groupoid G is ´etale[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 2 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 2 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 2 S. An element e 2 S is called an idempotent if ee = e. The set E of idempotents in S satisfies E = fs∗s : s 2 Sg. For all e; f 2 E, the greatest lower bound of fe; fg is ef. The set E is a commutative inverse semigroup. For all e 2 E and a 2 S, we have • 0 2 E, • e∗ = e, • a∗ea 2 E, • ae; ea ≤ a, and • a ≤ e =) a 2 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" := fb 2 S : a ≤ b for some a 2 Ag and A# := fb 2 S : b ≤ a for some a 2 Ag: For each a 2 S, we write a" := fag" and a# := fag#. For all e 2 E, we have e# ⊆ E. We say that A is down-directed if, for all a; b 2 A, there is c 2 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 2= 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.
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