Hausdorff tight groupoids generalised
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Abstract
We extend Exel’s ample tight groupoid construction to general locally compact étale groupoids in the Hausdorff case. Moreover, we show how inverse semigroups are represented in this way as ‘pseudobases’ of open bisections, thus yielding a duality which encompasses various extensions of the classic Stone duality.
Keywords
Stone duality Domain theory Local compactness Patch topology Étale groupoids Inverse semigroups Order structures Filters Cosets1 Introduction
1.1 Motivation
Exel’s original tight groupoid construction in [7] produces an ample (i.e. totally disconnected étale) groupoid from an inverse semigroup. Since then, this general construction has proved extremely successful in producing groupoid models for a vast array of C*algebras. The only caveat here is that C*algebras coming from ample groupoids inevitably have plenty of projections. Our ultimate goal is to produce combinatorial groupoid models even for projectionless C*algebras which means we must first find a nonample generalisation of the tight groupoid construction.
Remark
This task seems all the more urgent given that projectionless C*algebras like the JiangSu and JacelonRazak algebras have come to the fore in other contexts, namely the classification program for C*algebras. Whether these C*algebras have any groupoid models at all was only resolved recently in [4] and [1]. The hope would be that an even better picture of these C*algebras could be provided by more combinatorial groupoid models obtained from a generalised tight groupoid construction. One might even hope that important C*algebra properties (e.g. strong selfabsorption) could be detected at the inverse semigroup level.
1.2 Outline
In Exel’s original tight groupoid construction, the elements of the inverse semigroup get represented as compact open bisections in the resulting groupoid. To produce more general (potentially nonample) groupoids we should of course take the inverse semigroup elements to represent more general (potentially noncompact) open bisections. However, we still require some information about compactness in order to produce locally compact groupoids.
Remark
We will define several other abstract relations but, as above, in order to provide some intuition, we will say what each relation informally ‘represents’ for subsets of compact Hausdorff spaces.
At first we put the inverse semigroup structure to one side and focus just on the relation \(\prec \). In Sect. 1, we take \(\prec \) to be a binary relation on an arbitrary set P, from which we define cover relations \(\mathrel {\mathsf {D}}\) and \(\mathrel {\mathsf {C}}\). We then consider centred subsets and finish with a selection principle extending König’s Lemma in Lemma 1.17.
In Sect. 2, we move on to the topological part of our construction. With just the single weak interpolation assumption in (Shrinking), we are able to show that the tight spectrum always produces a locally compact space in Corollary 2.28. Moreover, this yields a duality between abstract and concrete ‘pseudobases’ of locally compact Hausdorff spaces, as we show in Proposition 2.44 and Theorem 2.45.
In Sect. 3, we extend this to inverse semigroups, showing how the the algebraic structure passes to the topological space. This yields a duality between pseudobasic inverse semigroups and étale pseudobases of locally compact Hausdorff étale groupoids, as shown in Theorem 3.23 and Proposition 3.26.
1.3 Background
We have taken inspiration from a number sources, notably domain theory (see [12] and [13]), set theory (see [16]), pointfree topology (see [22]) and its noncommutative generalisations (see [20, 23] and [14]). We also discovered that the earlier parts of our theory were developed independently in some of Exel’s unpublished notes, namely [9] and [10] (kindly provided to us by the author after presenting our work at the UFSC Operator Algebra Seminar in August 2018), as we mention at relevant points below.
Probably the best way of putting our construction in context is to view it as a generalisation of various Stone type dualities, as summarised in the diagram below. Note each duality is connected above to all those dualities it generalises and is expressed in the following form:
Author (Year) [Paper]
Abstract Structure
Concrete (Pseudo)basis
Topological Space/Groupoid
E.g. the regular \({}^{c\circ }\)\({\overline{\cup }}^\circ \)\(\cap \)bases below considered by De Vries refer to bases of regular open sets (i.e. \(O={\overline{O}}^\circ \)) that are closed under regular complements (i.e. \(O^{c\circ }\)), regular pairwise unions (i.e. \(\overline{O\cup N}^\circ \)) and pairwise intersections (i.e. \(O\cap N\)). Also LCLH stands for ‘locally compact locally Hausdorff’ (and for compact or Hausdorff spaces we remove the corresponding L).
Remark
Our construction can also be viewed as a generalization of a classical set theoretic construction of a Boolean algebra from a poset (see [16, Lemma III.4.8] and [3, §7]). Specifically, given any poset \((P,\le )\), the regular open sets in the Alexandroff topology form a Boolean algebra and, moreover, P is dense in the subalgebra it generates, which can then be identified with its (totally disconnected) Stone space. The topological part of our construction extends this from posets \((P,\le )\) to abstract pseudobases \((P,\prec )\) where \(\prec \) need not be reflexive and the resulting spaces are no longer necessarily totally disconnected.
One key difference we should point out is that, while the other dualities above are first order, this is no longer the case with the (generalised) tight groupoid construction. This is because we must often work with subsets rather than single elements. However, for the most part we can work with finite subsets so the theory could be expressed in terms of omitting types (as in [3, §5], which the present paper generalises), which makes it first order in a weak sense. In particular, the abstract pseudobases we deal with would still provide a natural way of interpreting locally compact spaces in forcing extensions (Wallman’s normal lattices were used for this purpose instead in [15]).
In contrast, the frames/locales/quantales appearing in pointfree topology and their noncommutative generalizations are undeniably second order, making vital use of infinite joins. Even the finite joins and the corresponding distributivity required in [18] and [2] restricts the freedom one has in making combinatorial constructions. Thus, while these various pointfree counterparts of étale groupoids are attractive from an abstract theory point of view, it is rather the (generalised) tight groupoid construction that is more suitable for creating specific examples.
Remark
A similar phenomenon occurs in set theory, where forcing is usually done with combinatorial posets rather than Boolean algebras, even though they produce the same extensions.
2 Preliminaries
Throughout we assume \(\prec \)is a binary relation on a setP.
First let us make some general notational conventions.
Definition 1.1
2.1 Cover relations
Definition 1.2
Note here it is important that we consider the elements of P to represent nonempty sets, i.e. P will not have a minimum 0 (i.e. satisfying \(0^\prec =P\)) except in trivial cases. Indeed, in Sect. 3 when we consider an inverse semigroup S with 0, we will take \(P=S{\setminus }\{0\}\). However, \({\mathcal {P}}(P)\) certainly has a minimum with respect to \( \subseteq \), namely the empty set \(\emptyset \), which we immediately see is also a minimum with respect to \(\prec \), \(\mathrel {\mathsf {D}}\) and \(\mathrel {\mathsf {C}}\) on \({\mathcal {P}}(P)\), i.e. \(\emptyset \prec Q\), \(\emptyset \mathrel {\mathsf {D}}Q\) and \(\emptyset \mathrel {\mathsf {C}}Q\), for all \(Q \subseteq P\).
Remark 1.3
Alternatively, one could assume that P always has a minimum 0 and then modify the definitions of \(\mathrel {\mathsf {D}}\) and \(\mathrel {\mathsf {C}}\) accordingly. However, we would then have to say ‘nonzero’ all the time, e.g. ‘for all nonzero p...’ or ‘there exists nonzero p...’, which can get somewhat tiresome.
Remark 1.4
In the classical case when \(\prec \) is a preorder \(\le \), \(\mathrel {\mathsf {D}}\) is essentially the same as the Lenz arrow relation in [18], originally from [19]. Equivalently, \(Q\mathrel {\mathsf {D}}R\) is saying that R is an ‘outer cover’ of \(Q^\ge \) in sense of [6]. There is also another more topological description of \(\mathrel {\mathsf {D}}\), which provides extra motivation for calling it the ‘dense cover relation’ (in fact the ‘dense’ terminology is already used for singleton sets in [7, Definition 11.10]). Specifically, \(Q\mathrel {\mathsf {D}}R\) means that \(Q^\ge \cap R^\ge \) is dense in \(Q^\ge \) in the Alexandroff topology, where the open sets are precisely the lower sets. Also, if \(R\le Q\) then, in set theoretic terminology, \(Q\mathrel {\mathsf {D}}R\) means ‘R is predense in \(Q^\ge \)’ – see [16, Definition III.3.58].
Proposition 1.5
Proof

(1.4) Consequently, \(Q\mathrel {\mathsf {D}}S\mathrel {\mathsf {D}}F\prec R\) implies \(Q\mathrel {\mathsf {D}}F\prec R\), so \(Q\mathrel {\mathsf {D}}S\mathrel {\mathsf {C}}R\) implies \(Q\mathrel {\mathsf {C}}R\). Also \(Q\mathrel {\mathsf {D}}F\prec R\) implies \(Q^\succ \succ F^\succ \subseteq R^{\succ \succ } \subseteq R^\succ \), i.e. \(Q\mathrel {\mathsf {C}}R\) implies \(Q\mathrel {\mathsf {D}}R\). Thus \(Q\mathrel {\mathsf {C}}R\mathrel {\mathsf {C}}S\) implies \(Q\mathrel {\mathsf {D}}R\mathrel {\mathsf {C}}S\) and hence \(Q\mathrel {\mathsf {C}}S\), i.e. \(\mathrel {\mathsf {C}}\) is also transitive. Finally note that if \(Q\mathrel {\mathsf {D}}R\) then, for all \(q\prec Q\), we have \(r\prec q\prec Q\) with \(r\prec R\). By transitivity, \(r\prec Q\) so we can further take \(r'\prec r\) (with \(r'\prec R\)), so \(r'\prec q\) and \(r'\in R^{\succ \succ }\), i.e. \(Q\mathrel {\mathsf {D}}R\) implies \(Q\mathrel {\mathsf {D}}R^\succ \).
 (1.5) Assume \(Q^\succ \succ Q'^\succ \) and \(R^\succ \succ R'^\succ \). For any \(p\in Q^\succ \cap R^\succ \), we have \(p'\prec p\) with \(p'\prec Q'\), as \(Q^\succ \succ Q'^\succ \). Then \(p'\prec p\prec R\) so \(p'\prec R\), by the transitivity of \(\prec \), and hence we have \(p''\prec p'\) with \(p''\prec R'\), as \(R^\succ \succ R'^\succ \). Thus \(p''\prec p'\prec Q'\) so \(p''\prec Q'\) and \(p''\prec p'\prec p\) so \(p''\prec p\), again by the transitivity of \(\prec \). Thus \(p''\in R'^\succ \cap Q'^\succ \) and, arguing as above, we can further obtain \(p'''\prec p''\) so \(p'''\in (R'^\succ \cap Q'^\succ )^\succ \). Theni.e. \((R^\succ \cap Q^\succ )^\succ \succ (R'^\succ \cap Q'^\succ )^\succ \), as required.$$\begin{aligned}(R^\succ \cap Q^\succ )^\succ \subseteq (R^\succ \cap Q^\succ )\succ (R'^\succ \cap Q'^\succ )^\succ ,\end{aligned}$$
 (1.6) Simply note that \(Q^\succ \succ Q'^\succ \) and \(R^\succ \succ R'^\succ \) implies$$\begin{aligned} (Q\cup R)^\succ =Q^\succ \cup R^\succ \succ Q'^\succ \cup R'^\succ =(Q'\cup R')^\succ .\end{aligned}$$

(1.7) Now \(Q\mathrel {\mathsf {D}}F\prec Q'\) and \(R\mathrel {\mathsf {D}}G\prec R'\) implies \(Q\cup R\mathrel {\mathsf {D}}F\cup G\prec Q'\cup R'\).
2.2 Round subsets
Other properties of \(\mathrel {\mathsf {C}}\) and \(\mathrel {\mathsf {D}}\) require an extra assumption.
Definition 1.6
Equivalently, \(R \subseteq P\) is round if iff R has no strictly minimal elements (where \(r\in R\) is strictly minimal if \(q\not \prec r\), for any \(q\in R\)). The ‘round’ terminology is common when talking about ideals, for example, in domain theory  see [13, Proposition 5.1.33] or [12, Proposition III4.3]. As long as P itself is round, \(P\mathrel {\mathsf {D}}P\) and \(\emptyset \) is the only (nonstrict) minimum for \(\mathrel {\mathsf {D}}\) and \(\mathrel {\mathsf {C}}\).
Proposition 1.7
Proof

(1.8) \(Q\prec \emptyset \) means \(Q \subseteq \emptyset ^\succ =\emptyset \) and hence \(Q=\emptyset \).

(1.9) \(Q\mathrel {\mathsf {D}}\emptyset \) means \(Q^\succ \succ \emptyset ^\succ =\emptyset \) and hence \(Q^\succ =\emptyset \), by (1.8). If P is round then this means \(Q=\emptyset \).

(1.10) \(Q\mathrel {\mathsf {C}}\emptyset \) means \(Q\mathrel {\mathsf {D}}F\prec \emptyset \) so \(F=\emptyset \), by (1.8), and hence \(Q=\emptyset \), by (1.9).
Proposition 1.8
Proof

(1.11) If P is round then \(Q \subseteq Q^{\succ \prec }\), for all \(Q \subseteq P\). Replacing Q with \(R^\succ \) yields \(R^\succ \subseteq R^{\succ \succ \prec } \subseteq R^{\succ \prec }\), as \(R^\succ \subseteq R^{\succ \succ }\) because \(\prec \) is transitive, i.e. \(R\mathrel {\mathsf {D}}R\). Thus \(Q \subseteq R\mathrel {\mathsf {D}}R\) implies \(Q\mathrel {\mathsf {D}}R\), by (1.1).

(1.12) By (1.11), \(F\prec R\) implies \(F\mathrel {\mathsf {D}}F\prec R\), which means \(F\mathrel {\mathsf {C}}R\).
2.3 Centred subsets
Definition 1.9
The term ‘centred’ comes from [16, Definition III.3.23]. It will be convenient to also define centred versions of the relations we have considered so far.
Definition 1.10
Lemma 1.11
Proof
By (1.10), we can apply this result with \(R=\emptyset \) to obtain the following corollary (note here that P has to be round, otherwise we could potentially have \(F=\emptyset \)).
Corollary 1.12
2.4 Frink filters
Definition 1.13
Remark 1.14
2.5 Disjoint subsets
Definition 1.15
Proposition 1.16
Proof
If we had \(Q\not \perp R\), then we would have \(q\in Q^\succ \cap R^\succ \). Then \(Q\mathrel {\mathsf {D}}Q'\) would yield \(q'\prec q\) with \(q'\prec Q'\) and hence \(q'\in Q'^\succ \cap R^\succ \), i.e. \(Q'\not \perp R\). \(\square \)
2.6 A selection principle
 1.
\(\Delta \) represents the finite families of open sets with nonempty intersection.
 2.
\(\Gamma \) represents a round collection of finite families whose unions have the FIP (i.e. finite subsets have nonempty intersection).
Lemma 1.17
Proof
Next, take disjoint subsets \(F^+,F^ \subseteq F\in \Gamma \). We claim that we can remove at least one of these subsets from F again without destroying the \(\Delta \)centred property of \(\Gamma \). If not then we could find finite \(\Phi ^+ \subseteq \Gamma \) and \(\Phi ^ \subseteq \Gamma \) such that \(\Phi ^+\cup \{F{\setminus } F^+\}\) and \(\Phi ^\cup \{F{\setminus } F^\}\) are not \(\Delta \)centred. This means that, for any \(D\in \Delta \) such that \(D\cap G\ne \emptyset \), for all \(G\in \Phi ^+\), we have \(D\cap F \subseteq F^+\). Likewise, for any \(D\in \Delta \) such that \(D\cap G\ne \emptyset \), for all \(G\in \Phi ^\), we have \(D\cap F \subseteq F^\). Thus, for any \(D\in \Delta \) such that \(D\cap G\ne \emptyset \), for all \(G\in \Phi ^+\cup \Phi ^\), we have \(D\cap F \subseteq F^+\cap F^=\emptyset \), contradicting the \(\Delta \)centred property of \(\Gamma \).
Finally, let \(R=\bigcup _\lambda G_\lambda \), which certainly satisfies (\(\Gamma \)Selector). Also R is round, as \((G_\lambda )_{\lambda \in \Lambda }\) consists of \(\prec \)round singletons. Moreover, R satisfies (\(\Delta \)FIP), as \((G_\lambda )_{\lambda \in \Lambda }\) consists of \(\Delta \)centred singletons. \(\square \)
3 Pseudobases
3.1 The shrinking condition
From now on we will focus on transitive relations \(\prec \) that make P round. However, there is one more condition that we need to be able to define general locally compact Hausdorff spaces from \((P,\prec )\). Specifically, we need a condition which represents the ability to ‘shrink’ a cover to another cover, each element of which is compactly contained in some element of the original cover.
Definition 2.1
Remark 2.2
However, it would seem that (Shrinking) is more appropriate for our work than (Interpolation), as (Shrinking) applies to arbitrary (pseudo)bases of locally compact Hausdorff spaces, even when they are not closed under taking finite unions. The extra freedom coming from (Shrinking) will also be convenient for constructing specific examples. In any case, (Shrinking) suffices to guarantee that we have a large supply of tight subsets, as we will soon see.
First we note (Shrinking) extends to \(\mathrel {\mathsf {C}}\), allowing us to replace \(\mathrel {\mathsf {D}}\) with \(\mathrel {\mathsf {C}}\) in (1.4).
Proposition 2.3
Proof
3.2 Tight subsets
From now on we assume \((P,\prec )\) is an abstract pseudobasis
Again recall that we are thinking of elements of P as open subsets of a topological space. It should then be possible to recover the points of the space from their open neighbourhoods. The key thing to note here is that no neighborhood of a point x can be covered by subsets which do not contain x. Moreover, in locally compact spaces every neighborhood of x compactly contains a smaller neighbourhood of x. Thus the neighbourhoods of x in P are ‘tight’ in the following sense.
Definition 2.4
We call \(T \subseteq P\)tight if Open image in new window .
Proposition 2.5
If \(T \subseteq P\) is tight then T is a Frink filter.
Proof
The \(\Rightarrow \) part of (Frink Filter) is immediate because T is round. Conversely, if \(T\mathrel {{\widehat{\prec }}}v\prec u\) then \(T\mathrel {{\widehat{\mathrel {\mathsf {D}}}}}v\prec u\) so \(T\mathrel {{\widehat{\mathrel {\mathsf {C}}}}}u\). Thus \(u\in T\), as Open image in new window . \(\square \)
In particular, any tight T is upwards closed and thus \(T=T^\prec \), as T is also round.
Remark 2.6
If \((P,\prec )=(S{\setminus }\{0\},\le )\), for some \(\wedge \)semilattice S with minimum 0, then every tight subset is a filter, by Remark 1.14 and Proposition 2.5. In this case, our tight subsets are precisely the tight filters defined in [6]. This becomes clear from the following general characterisation of tight subsets, as (Tight’) below extends [6, (2.10)].
Proposition 2.7
Proof
If T is round then, for any \(t\in T\), we have \(s\in T\) with \(s\prec t\) and hence \({\widehat{s}}\cap \emptyset ^\perp \mathrel {\mathsf {C}}t\). If Open image in new window and we have finite \(F \subseteq T\) and \(G\prec P{\setminus } T\) with \({\widehat{F}}\cap G^\perp \mathrel {\mathsf {C}}H\) then we have finite I with \({\widehat{F}}\cap G^\perp \mathrel {\mathsf {D}}I\prec H\) and hence \({\widehat{F}}\mathrel {\mathsf {D}}G\cup I\). If we had \(H\cap T=\emptyset \) then this would imply \({\widehat{F}}\mathrel {\mathsf {D}}G\cup I\prec P{\setminus } T\) and hence \(T\mathrel {\widehat{\mathsf {C}}}P{\setminus } T\), a contradiction, so \(H\cap T\ne \emptyset \). Thus if T is tight then T satisfies (Tight’). Also (Tight’) implies (Tight), as the latter is just the former restricted to singleton H.
Now say T satisfies (Tight). For any \(t\in T\), we have finite \(F \subseteq T\) and \(G\prec P{\setminus } T\) with \({\widehat{F}}\cap G^\perp \mathrel {\mathsf {C}}t\). By (\(\mathsf {C}\)Shrinking), \({\widehat{F}}\cap G^\perp \mathrel {\mathsf {C}}t^\succ \) so we have finite \(H,I \subseteq P\) with \({\widehat{F}}\cap G^\perp \mathrel {\mathsf {D}}I\prec H\prec t\) and hence \({\widehat{F}}\cap (G\cup I)^\perp =\emptyset \mathrel {\mathsf {C}}p\), for all \(p\in P\). Then \(H \subseteq P{\setminus } T\) would imply \(G\cup I\prec P{\setminus } T\) and hence \(T=P\), by (Tight), contradicting our assumption. Thus \(H\cap T\ne \emptyset \), which shows that T is round.
Still assuming T satisfies (Tight), say we had \(T\mathrel {\widehat{\mathsf {C}}}P{\setminus } T\), which means we have finite \(F \subseteq T\) and \(G\prec P{\setminus } T\) with \({\widehat{F}}\mathrel {\mathsf {D}}G\). Then \({\widehat{F}}\cap G^\perp =\emptyset \mathrel {\mathsf {C}}p\), for all \(p\in P\), so \(T=P\), by (Tight), again contradicting our assumption. Thus Open image in new window , showing that (Tight) does indeed imply that T is tight. \(\square \)
Incidentally, P itself will only be tight in trivial cases. At the other extreme, it is more common for \(\emptyset \) to be tight, although we will specifically omit \(\emptyset \) from the tight spectrum considered in the next section (in order to be able to obtain locally compact rather than just compact spaces).
Proposition 2.8
 (1)
P is tight iff P is centred, in which case P is the only tight subset of P.
 (2)
\(\emptyset \) is tight iff Open image in new window .
Proof

(1) By definition, P is tight iff Open image in new window . This means Open image in new window , for all finite \(F \subseteq P\), which is just saying that P is centred. In particular, this implies \(P\mathrel {\mathsf {D}}p\) and hence \(P\mathrel {\mathsf {C}}p\), for any \(p\in P\). Thus \(T\mathrel {\widehat{\mathsf {C}}}P{\setminus } T\) whenever \(T\ne P\) so there can be no other tight subsets.

(2) By definition, \(\emptyset \) is tight iff Open image in new window , which means Open image in new window .
Proposition 2.9
Proof
 (2.2) Any round extension of U must contain some \(a\prec P{\setminus } U\). So if the right side of (2.2) holds then such an extension could not be centred, which implies that U must be maximal. Conversely, say the right side of (2.2) fails, so we have some \(a\prec P{\setminus } U\) with \({\widehat{F}}\not \perp a\), for all finite \(F \subseteq U\), i.e. \(U\cup \{a\}\) is centred. By (Shrinking), we have a sequence \((F_n)\) of finite subsets of P such that, for all n,In particular, for all n, \(a\mathrel {\mathsf {D}}F_n\) which implies that \(U\cup \{f\}\) is centred, for some \(f\in F_n\), by Corollary 1.12 (with \(Q=U\cup \{a\}\) and \(F=F_n\)). Thus we can apply Lemma 1.17, taking \((F_n)\) for \(\Gamma \) and setting$$\begin{aligned}a\mathrel {\mathsf {C}}F_{n+1}\prec F_n\prec P{\setminus } U.\end{aligned}$$to obtain round T such that \(U\cup T\) is centred and \(T\cap F_n\ne \emptyset \), for all n (actually \(F'_n=\{f\in F_n:U\cup \{f\}\text { is centred}\}\) defines a finitely branching \(\omega \)tree so in this case we could also obtain T from a simple application of König’s lemma  see [16, Lemma III.5.6]). Thus \(U\cup T^\prec \) is a round centred subset containing some element of \(P{\setminus } U\) so U could not have been maximal.$$\begin{aligned} \Delta =\{D \subseteq P:D\text { is finite and }U\cup D\text { is centred}\}, \end{aligned}$$

(2.3) We immediately see that (2.3) implies (2.2). Conversely, if (2.2) holds and we are given finite \(G\prec P{\setminus } U\) then, for each \(g\in G\), we have finite \(F_g \subseteq U\) with \({\widehat{F}}_g\perp g\). Taking \(F=\bigcup _{g\in G}F_g\), we then have \({\widehat{F}}\perp G\).

(2.3)\(\Rightarrow \)(2.4) If (2.3) holds and we are given \(G\prec P{\setminus } U\) then we have finite \(F \subseteq U\) with \({\widehat{F}}\perp G\). For any other finite \(F' \subseteq U\), we have \(\emptyset \ne \widehat{F\cup F'}\perp G\), as U is centred, so Open image in new window .

(2.4) Just note that the right side of (2.4) means Open image in new window .
The following result shows that we can even be more selective about our tight extensions. For example, we might want to find a tight extension of some round R but still avoid another given subset S. The following result (in the \(Q=\emptyset \) case) shows that this can be done as long as Open image in new window .
Remark 2.10
Consequently, this result is essentially an extension of Birkhoff’s prime ideal theorem for distributive lattices (see [12, Lemma I3.20]). Indeed, if \((P,\prec )=(L{\setminus }\{0\},\le )\), for some separative distributive lattice L with minimum 0, then the tight subsets are precisely the prime filters. In this case, Open image in new window is saying that the filter generated by R is disjoint from the ideal generated by S. Birkhoff’s theorem says that R therefore extends to a prime filter disjoint from S.
Theorem 2.11
Proof
3.3 The tight spectrum
Recall our standing assumption that P is an abstract pseudobasis
Definition 2.12
Remark 2.13
If we allowed \(\emptyset \) to be part of the tight spectrum then we would always obtain compact rather than locally compact spaces. A more serious issue arises for inverse semigroups, where we need to avoid the empty subset to ensure that multiplying tight subsets yields a groupoid operation.
Remark 2.14
For anyone familiar with Stone duality or pointfree topology, it might be tempting to consider the weaker topology on tight subsets generated just by the sets \(O_f=\{T\in {\mathcal {T}}(P):f\in T\}\). The problem is that, for general abstract pseudobases, the resulting space may not be Hausdorff or even \(T_1\), although it will still be locally compact and \(T_0\). A standard way of strengthening such a topology to make it Hausdorff is to consider the patch topology as in Proposition 2.37 below. Indeed, combined with Proposition 2.44, this shows that the resulting patch topology is precisely the topology obtained by adding the sets \(O^g=\{T\in {\mathcal {T}}(P):g\mathrel {\mathsf {C}}P{\setminus } T\}\) to the subbasis. This answers a question posed by Gilles de Castro.
Remark 2.15
Note \(G\mathrel {\mathsf {C}}P{\setminus } T\) implies \(G \subseteq P{\setminus } T\), but not conversely. That is unless \(\prec \) is reflexive and hence a preorder—then \(G\mathrel {\mathsf {C}}P{\setminus } T\) is indeed the same as \(G \subseteq P{\setminus } T\) and the topology in this case could be viewed as coming from the product topology on the characteristic functions of the tight subsets. Thus when \((P,\prec )=(S{\setminus }\{0\},\le )\), for some \(\wedge \)semilattice S with minimum 0, our tight spectrum agrees with Exel’s original tight spectrum—see [6].
Remark 2.16
Now we consider topological properties of the tight spectrum.
Proposition 2.17
The tight spectrum is Hausdorff.
Proof
Take distinct \(T,U\in {\mathcal {T}}(P)\). Thus we must have \(t\in T{\setminus } U\) (or vice versa). As T is round, we have \(s\in T\) with \(s\prec t\), which means \(T\in O_s\) and \(U\in O^s\). If \(s\in V\in {\mathcal {T}}(P)\) then Open image in new window and hence Open image in new window so \(O_s\cap O^s=\emptyset \). \(\square \)
Note the above proof used the fact that \(O^s_s=\emptyset \). More generally, for any finite \(F,G \subseteq P\), we can characterize when \(O_F^G=\emptyset \) as follows.
Proposition 2.18
Proof
Assume \(G\mathrel {\mathsf {C}}H\) and Open image in new window . By (\(\mathsf {C}\)Shrinking), we have finite \(G'\) with \(G\mathrel {\mathsf {C}}G'\prec H\) and hence Open image in new window . This means we have p with \(F\succ p\perp G'\). As P is round, we have a sequence \(p=p_1\succ p_2\succ \ldots \). By the Kuratowski–Zorn lemma, this sequence extends to some maximal round centred T, necessarily with \(F \subseteq T\) and \(T\cap G'=\emptyset \). By (2.4), T is tight so \(T\in O_F^G\), proving the ‘only if’ part.
Conversely, if \(T\in O_F^G\) then \(F \subseteq T\) and \(G\mathrel {\mathsf {C}}P{\setminus } T\). As T is tight, Open image in new window so Open image in new window , proving the ‘if’ part. \(\square \)
This has several important corollaries, e.g. we can generalise [7, Theorem 12.9] which says that ultrafilters are dense in the tight spectrum of a \(\wedge \)semilattice.
Corollary 2.19
The maximal round centred subsets of P are dense in \({\mathcal {T}}(P)\).
Proof
Any nonempty open set contains some nonempty \(O_F^G\). Now just note that the \(T\in O_F^G\) in the proof of Proposition 2.18 is a maximal round centred subset. \(\square \)
Corollary 2.20
Proof
If \(p\perp q\) then no centred set can contain both p and q so \(O_p\cap O_q=\emptyset \). Conversely, if \(O_{p,q}^\emptyset =O_p\cap O_q=\emptyset \) then, as \(\emptyset \mathrel {\mathsf {C}}\emptyset \), we must have \(\widehat{\{p,q\}}\mathrel {\mathsf {C}}\emptyset \), by Proposition 2.18. Thus \(\widehat{\{p,q\}}=\emptyset \), by (1.10), i.e. \(p\perp q\). \(\square \)
Corollary 2.21
If \({\widehat{F}}\mathrel {\mathsf {D}}G\) then \(O_F^G=\emptyset \).
Proof
If \({\widehat{F}}\mathrel {\mathsf {D}}G\mathrel {\mathsf {C}}H\) then \({\widehat{F}}\mathrel {\mathsf {C}}H\), by (1.4), so \(O_F^G=\emptyset \), by Proposition 2.18. \(\square \)
Conversely, if \(O_F^G=\emptyset \) and P is a poset then G itself is a \(\mathrel {\mathsf {C}}\)/\(\mathrel {\mathsf {D}}\)cover of G so \({\widehat{F}}\mathrel {\mathsf {D}}G\), by Proposition 2.18.
Example 2.22
This no longer holds when P is not a poset. For example, let \(P={\mathbb {N}}\cup \{a\}\) and let \(\prec \) be \(=\) on \({\mathbb {N}}\) while \(n\prec a\) iff \(n\ne 2\) and \(a\not \prec a\). Then \(2\perp a\) and, in particular, Open image in new window even though a has no \(\mathrel {\mathsf {C}}\)cover and hence \(O^a_{2}=\emptyset \).
We can now show that \(\mathrel {\mathsf {D}}\) has the desired representation in \({\mathcal {T}}(P)\).
Proposition 2.23
Proof
If Open image in new window then we have \(p\in P\) with \(F\succ p\perp Q\), which means \(O_F\supseteq O_p\) and \(O_p\cap \bigcup _{q\in Q}O_q=\emptyset \), by (2.7). By (2.8), \(\emptyset \ne O_p \subseteq O_F{\setminus }\overline{\bigcup _{q\in Q}O_q}\) so \(O_F\not \subseteq \overline{\bigcup _{q\in Q}O_q}\).
Conversely, if \(O_F\not \subseteq \overline{\bigcup _{q\in Q}O_q}\) then we have \(T\in O^G_H \subseteq O_F{\setminus }\overline{\bigcup _{q\in Q}O_q}\), for some finite \(G,H \subseteq P\). As \(G\mathrel {\mathsf {C}}P{\setminus } T\), (\(\mathsf {C}\)Shrinking) yields I with \(G\mathrel {\mathsf {C}}I\prec P{\setminus } T\). As T is tight, \(F\cup H \subseteq T\) and \(I\prec P{\setminus } T\), we have some \(p\in P\) with \(F\cup H\succ p\perp I\). Thus \(O_p \subseteq O^G_H\) and hence \(O_p\cap \bigcup _{q\in Q}O_q=\emptyset \) so \(p\perp Q\), by (2.7). As we also have \(F\succ p\), this shows that Open image in new window . \(\square \)
Corollary 2.24
Proof
Note \(R\mathrel {\mathsf {D}}Q\) iff \(r\mathrel {\mathsf {D}}Q\), for all \(r\in R\). By (2.9), this is the same as saying \(O_r \subseteq \overline{\bigcup _{q\in Q}O_q}\), for all \(r\in R\), i.e. \(\bigcup _{r\in R}O_r \subseteq \overline{\bigcup _{q\in Q}O_q}\). \(\square \)
Next we wish to do the same for \(\mathrel {\mathsf {C}}\). First we need the following.
Proposition 2.25
Proof
Note \(T\notin \overline{O_F}\) iff we have a basic neighbourhood of T disjoint from \(O_F\), i.e. iff we have finite \(F' \subseteq T\) and \(G\mathrel {\mathsf {C}}P{\setminus } T\) with \(O_{F\cup F'}^G=O_F\cap O_{F'}^G=\emptyset \). By (2.6), this implies \(\widehat{F'\cup F}\mathrel {\mathsf {C}}P{\setminus } T\) and hence \(T\cup F\mathrel {\widehat{\mathsf {C}}}P{\setminus } T\).
Conversely, if \(T\cup F\mathrel {\widehat{\mathsf {C}}}P{\setminus } T\) then we have finite \(F' \subseteq T\) with \(\widehat{F'\cup F}\mathrel {\mathsf {C}}P{\setminus } T\) and hence we have finite G with \(\widehat{F'\cup F}\mathrel {\mathsf {D}}G\prec P{\setminus } T\). So \(G\mathrel {\mathsf {C}}P{\setminus } T\), by (1.12), and \(O_F\cap O_{F'}^G=O_{F\cup F'}^G=\emptyset \), by Corollary 2.21. Thus \(T\in O_{F'}^G\) and hence \(T\notin \overline{O_F}\). \(\square \)
Theorem 2.26
Proof
Assume F is finite and \({\widehat{F}}\mathrel {\mathsf {C}}Q\). Then, for any \(T\in {\mathcal {T}}(P){\setminus }\bigcup _{q\in Q}O_q\), we have \({\widehat{F}}\mathrel {\mathsf {C}}Q \subseteq P{\setminus } T\) so \(T\notin \overline{O_F}\), by Proposition 2.25. Thus \(\overline{O_F} \subseteq \bigcup _{q\in Q}O_q\).
To prove that \(\overline{O_F}\) is compact, it suffices to show that every subbasic open cover has a finite subcover, by the Alexander subbasis theorem. So assume we have some \(S \subseteq P\) and a collection \({\mathcal {G}}\) of finite subsets of P such that no finite subcollection of \((O_s)_{s\in S}\) and \((O^G)_{G\in {\mathcal {G}}}\) covers \(\overline{O_F}\) (in particular we must have \(O_F\ne \emptyset \) and hence \({\widehat{F}}\ne \emptyset \)). We can also assume that \({\mathcal {G}}\) contains some G with \(G\mathrel {\mathsf {C}}H\), for some finite H. Indeed, we have G with \({\widehat{F}}\mathrel {\mathsf {C}}G\prec Q\) and hence \(\emptyset \ne G\prec H\ne \emptyset \) for some finite \(H \subseteq Q\). Thus \(\overline{O_F}\cap O^G=O_F^G=\emptyset \), which means we could add G to \({\mathcal {G}}\) and there would still be no finite subcollection covering \(\overline{O_F}\).
Applying Lemma 1.17 we obtain \(R \subseteq P\) such that \(H\cap R\ne \emptyset \), for all \(H\in \Gamma \), and \(D\in \Delta \), for all finite \(D \subseteq R\), which means Open image in new window . By Theorem 2.11, we have tight \(T\supseteq R\) with Open image in new window . Thus \(T\ne \emptyset \), as \(\Gamma \ne \emptyset \), and \(T\in \overline{O_F}\), by (2.10). Moreover \(T\notin \bigcup _{s\in S}O_s\), as \(S \subseteq P{\setminus } T\), and \(T\notin \bigcup _{G\in {\mathcal {G}}}O^G\), as \(H\cap T\ne \emptyset \), for all H with \(G\mathrel {\mathsf {C}}H\), for some \(G\in {\mathcal {G}}\). Thus \(T\in \overline{O_F}{\setminus }(\bigcup _{s\in S}O_s\cup \bigcup _{G\in {\mathcal {G}}}O^G)\) and hence the entirety of \((O_s)_{s\in S}\) and \((O^G)_{G\in {\mathcal {G}}}\) does not cover \(\overline{O_F}\) either. This shows that \(\overline{O_F}\) is indeed compact and hence \(O_F\Subset \bigcup _{q\in Q}O_q\).
On the other hand, if Open image in new window then \(\overline{O_F}\) is not even compact so Open image in new window . To see this, note that \((O_p)_{p\in P^\succ }\) covers the entire tight spectrum, as each \(T\in {\mathcal {T}}(P)\) is nonempty and round. However, for any finite \(G\prec P\), Open image in new window yields Open image in new window so we have f with \(F\succ f\perp G\). By (2.8), we have \(T\in O_f \subseteq O_F\) and hence \(T\notin \bigcup _{g\in G}O_g\), by (2.7). As G was arbitrary, this shows \((O_p)_{p\in P^\succ }\) has no finite subcover of \(O_F\). \(\square \)
Corollary 2.27
Proof
Recall that a Hausdorff space is locally compact iff each point has a compact neighbourhood (which implies that each point actually has a neighbourhood base of compact sets and is thus consistent with Definition 2.30 below).
Corollary 2.28
The tight spectrum is locally compact.
Proof
Any \(T\in {\mathcal {T}}(P)\) is nonempty and round, so we have \(s,t\in T\) with \(s\prec t\). By (1.12), \(s\mathrel {\mathsf {C}}t\) so \(T\in O_s\Subset O_t\), by Theorem 2.26. In particular, T has a compact neighbourhood so, as T was arbitrary, \({\mathcal {T}}(P)\) is locally compact. \(\square \)
For \({\mathcal {T}}(P)\) to be locally compact as above, (Shrinking) is crucial, as the following example shows (this answers a question posed by Exel).
Example 2.29
3.4 Patch topologies
We are primarily concerned with pseudobases (to be introduced in the next section) in locally compact Hausdorff spaces. However, it will be useful to first consider more general pseudosubbases and how they relate to more general stably locally compact \(T_0\) spaces via the patch construction.
Definition 2.30
 (1)
locally compact if each point has a neighbourhood base of compact sets.
 (2)coherent if X is locally compact and \({\mathcal {C}}(X)\) is closed under \(\cap \), i.e.
 (3)stably locally compact if X is coherent and, for \(O\in {\mathcal {O}}(X)\) and \({\mathcal {C}} \subseteq {\mathcal {C}}(X)\)$$\begin{aligned} \bigcap {\mathcal {C}} \subseteq O\quad \Rightarrow \quad \exists \text { finite }F \subseteq {\mathcal {C}}\ (\bigcap F \subseteq O). \end{aligned}$$(WellFiltered)
Remark 2.31
 (1)
locally compact if \(\{O\in {\mathcal {O}}(X):x\in O\}\) is \(\Subset \)round, for each \(x\in X\).
 (2)stably locally compact if, for \(O,O',N,N'\in {\mathcal {O}}(X)\) and \(\Subset \)round \({\mathcal {O}} \subseteq {\mathcal {O}}(X)\),$$\begin{aligned} O'\Subset O\ \text {and}\ N'\Subset N\quad&\Rightarrow \quad O'\cap N'\Subset O\cap N,\quad \text {and}\\ \bigcap {\mathcal {O}} \subseteq O\quad&\Rightarrow \quad \exists \text { finite }F \subseteq {\mathcal {O}}\ (\bigcap F \subseteq O). \end{aligned}$$
In a stably locally compact space X, both \({\mathcal {C}}(X)\) and \(\Subset \) behave like they would in a Hausdorff space. Indeed, in this case \({\mathcal {C}}(X)\) and \(\Subset \) are stable under the patch construction, which makes X Hausdorff, as long as X was originally \(T_0\).
Proposition 2.32
Proof
First note that the patch topology of any locally compact \(T_0\) space is Hausdorff, by [13, Proposition 9.1.12 and Lemma 9.1.31].
For any \(C,D\in {\mathcal {C}}(X)\), \(C\cap D\in {\mathcal {C}}(X)\), as X is coherent. Thus \(C^\mathrm {patch}\), the patch topology coming from the subspace topology of C, coincides with the subspace topology coming from \(X^\mathrm {patch}\). By [13, Proposition 9.1.27], \(C^\mathrm {patch}\) is compact so C is compact in \(X^\mathrm {patch}\) and hence \(C\in {\mathcal {C}}(X^\mathrm {patch})\), as X is Hausdorff.
As X is locally compact, every \(x\in X\) has a neighbourhood in \({\mathcal {C}}(X) \subseteq {\mathcal {C}}(X^\mathrm {patch})\). As \(X^\mathrm {patch}\) is Hausdorff, it follows that \(X^\mathrm {patch}\) is also locally compact.
Now take any \(O,N\in {\mathcal {O}}(X)\). If \(O\Subset N\) in \(X^\mathrm {patch}\) then this remains true in X because any compact subset of \(X^\mathrm {patch}\) is certainly compact in the coarser original topology of X. Conversely, if \(O\Subset N\) in X then we have compact C in X with \(O \subseteq C\subseteq N\). But the saturation of C is also compact, by [13, Proposition 4.4.14], and is still contained in the open set N. Thus we may take \(C\in {\mathcal {C}}(X) \subseteq {\mathcal {C}}(X^\mathrm {patch})\) and hence \(O\Subset N\) in \(X^\mathrm {patch}\). \(\square \)
Conversely, to produce general stably locally compact \(T_0\) spaces from locally compact Hausdorff spaces, we consider pseudosubbases.
Definition 2.33
Remark 2.34
Recall that a subbasisS of a space X is a collection of open sets that generates the topology of X, i.e. such that every element of \({\mathcal {O}}(X)\) is a union of finite intersections of elements of S. In particular, every subbasis of a locally compact \(T_0\) space is a pseudosubbasis. Conversely, if X has a pseudosubbasis then X must be both locally compact and \(T_0\).
Remark 2.35
By (Separating), any pseudosubbasis P must cover all except possibly one point of X, i.e. \(X{\setminus }\bigcup P\le 1\). Taking \(F=\emptyset \) in (\(\cap \)PointRound) we see that, for any \(x\in X=\bigcap \emptyset \), we have finite \(G \subseteq P\) with \(x\in \bigcap G\Subset X\). If X is compact then we can always take \(G=\emptyset \) too but otherwise G must be nonempty, i.e. if X is not compact then P must truly cover X.
Proposition 2.36
Any pseudosubbasis of a stably locally compact space X is a pseudosubbasis of \(X^\mathrm {patch}\).
Proof
Immediate because \(\Subset \) is the same on \({\mathcal {O}}(X)\) and \({\mathcal {O}}(X^\mathrm {patch})\), by (2.12). \(\square \)
Proposition 2.37
If P is a pseudosubbasis of a locally compact Hausdorff space X then \(X_P\) is a stably locally compact \(T_0\) space such that \(X=X_P^\mathrm {patch}\).
Proof
First replace P with its closure under finite intersections. Note P is still a pseudosubbasis of X and now also a basis of \(X_P\).
We claim that \({\mathcal {C}}(X_P) \subseteq {\mathcal {C}}(X)\). To see this, take any \(C\in {\mathcal {C}}(X_P)\) and \(x\in X{\setminus } C\). As C is saturated in \(X_P\), we have \(O\in {\mathcal {O}}(X_P)\) with \(x\notin O\supseteq C\). For every \(c\in C\), we have \(N\in P\) with \(c\in N \subseteq O\). As C is compact in \(X_P\), we thus have finite \(F \subseteq P\) with \(C \subseteq \bigcup F \subseteq O\). As x was arbitrary, this shows that \(C=\bigcap _{F\in \Gamma }\bigcup F\), where \(\Gamma \) is the collection of all finite \(F \subseteq P\) with \(C \subseteq \bigcup F\). Again by compactness in \(X_P\) and (\(\cap \)PointRound), \(\Gamma \) is \(\Subset \)round (where \(\Subset \) is defined from the original topology of X), i.e. for all \(F\in \Gamma \), we have \(G\in \Gamma \) with \(G \subseteq F^\Supset \). As X is Hausdorff this means \(C=\bigcap _{F\in \Gamma }\bigcup F=\bigcap _{F\in \Gamma }\overline{\bigcup F}\), where the closure is taken in X. Thus C is an intersection of compact subsets in the Hausdorff space X so \(C\in {\mathcal {C}}(X)\).
Thus \(X_P^\mathrm {patch}\) is Hausdorff, by [13, Proposition 9.1.12 and Lemma 9.1.31]. As \({\mathcal {C}}(X_P) \subseteq {\mathcal {C}}(X)\), the identity \(\mathbf {id}:X\rightarrow X_P^\mathrm {patch}\) is continuous. For any \(x\in X\), (\(\cap \)PointRound) yields \(O,N\in P\) with \(x\in N\Subset O\). As \({\overline{N}}\) is compact and \(X_P^\mathrm {patch}\) is Hausdorff, \(\mathbf {id}\) takes closed subsets of \({\overline{N}}\) to closed subsets, i.e. \(\mathbf {id}\) is a homeomorphism on \({\overline{N}}\) and hence on N. As N is also open in \(X_P^\mathrm {patch}\), \(\mathbf {id}\) is a local homeomorphism. But \(\mathbf {id}\) is clearly bijective and thus a global homeomorphism. \(\square \)
Remark 2.38
3.5 Concrete pseudobases
We have just seen in Proposition 2.37 that the topology of any locally compact Hausdorff space X can be recovered from any pseudosubbasis. However, this requires that we already know the points of X. What we would like to do is recover both the points and the topology from the relational structure \((P,\Subset )\), for suitable \(P \subseteq {\mathcal {O}}(X)\). Pseudosubbases are too general for this as the following elementary examples show.
Example 2.39
This leads us to consider the following stronger structures. Note (Dense) is the key extra condition distinguishing pseudobases from pseudosubbases.
Definition 2.40
Our primary goal in this section is to exhibit a duality between abstract and concrete pseudobases. First, however, we make some general observations.
To begin with, pseudobases are indeed generalisations of bases, at least when X is locally compact and \(T_0\). Indeed, bases of locally compact spaces satisfy (PointRound), while bases of \(T_0\) spaces satisfy (Separating) and, by definition, bases of arbitrary spaces satisfy (Cover) and (Dense).
Example 2.41
Consider the onepoint compactification \(X={\mathbb {N}}\cup \{\infty \}\) of \({\mathbb {N}}\) and let \(P=\{\{n\}:n\in {\mathbb {N}}\}\cup \{X\}\). It is immediately verified that P is a pseudobasis but not a basis, as it does not contain a neighbourhood base of the point \(\infty \).
Remark 2.42
Remark 2.43
Now for the promised duality between abstract and concrete pseudobases. First we represent abstract pseudobases as concrete pseudobases.
Proposition 2.44
If \((P,\prec )\) is an abstract pseudobasis then \((O_p)_{p\in P}\) is a concrete pseudobasis of \({\mathcal {T}}(P)\).
Proof

(Cover) As every \(T\in {\mathcal {T}}(P)\) is nonempty, we have some \(t\in T\) and hence \(T\in O_t\).

(PointRound) If \(T\in O_t\) then \(t\in T\) so, as T is round, we have \(s\in T\) with \(s\prec t\) and hence \(T\in O_s\Subset O_t\), by (1.12) and Theorem 2.26.

(Dense) If \(T\in O_F^G\) then \(F \subseteq T\) and \(G\mathrel {\mathsf {C}}S{\setminus } T\). By (\(\mathsf {C}\)Shrinking), we have H with \(G\mathrel {\mathsf {C}}H\prec P{\setminus } T\). By (2.1), we have p with \(F\succ p\perp H\) so \(O_p \subseteq O_F^G\). As the \((O_F^G)\) form a basis for \({\mathcal {T}}(P)\), this shows that any nonempty open \(O \subseteq {\mathcal {T}}(P)\) contains some \(O_p\), which is nonempty by (2.8).

(Separating) If \(T,U\in {\mathcal {T}}(P)\) are distinct then we have some \(t\in T{\setminus } U\) or \(t\in U{\setminus } T\), i.e. \(U\notin O_t\ni T\) or \(T\notin O_t\ni U\) so \(O_t\) distinguishes T from U.
Next we show how concrete pseudobases yield abstract pseudobases. Note the following extends [3, Theorem 8.4] to non0dimensional spaces.
Theorem 2.45
Proof

(2.15) If \((\bigcup Q)\cap (\bigcup R)=\emptyset \) then certainly \(Q^\succ \cap R^\succ \subseteq Q^\supseteq \cap R^\supseteq =\emptyset \). If \((\bigcup Q)\cap (\bigcup R)\ne \emptyset \) then we must have \(O\cap N\ne \emptyset \), for some \(O\in Q\) and \(N\in R\). Then (Dense) and (PointRound) yields \(M\in P\) with \(M\Subset O\cap N\) so \(O\not \perp N\).

(2.16) If Open image in new window then we have \(O\in Q^\succ \) with \(R\perp O\). Thus \(O \subseteq \bigcup Q\) and \(O\cap \bigcup R=\emptyset =O\cap \overline{\bigcup R}\) so \(\bigcup Q\not \subseteq \overline{\bigcup R}\). Conversely, if \(\bigcup Q\not \subseteq \overline{\bigcup R}\) then \(N{\setminus }\overline{\bigcup R}\ne \emptyset \), for some \(N\in Q\). Then (Dense) and (PointRound) yields \(O\in P\) with \(O\Subset N{\setminus }\bigcup R\) so Open image in new window .

(2.17) If G is Hausdorff and \(\bigcup Q\Subset \bigcup R\) then \(\overline{\bigcup Q}\) is compact and \(\overline{\bigcup Q} \subseteq \bigcup R\). Thus, for each \(g\in \overline{\bigcup Q}\), we have \(O\in R\) with \(g\in O\). Then (PointRound) yields \(N\in P\) with \(g\in N\Subset O\). As \(\overline{\bigcup Q}\) is compact we can therefore cover it with finite \(F \subseteq O^\succ \subseteq R^\succ \) so \(Q\mathrel {\mathsf {D}}F\prec R\), i.e. \(Q\mathrel {\mathsf {C}}R\). Conversely, if \(Q\mathrel {\mathsf {D}}F\prec R\) then \(\overline{\bigcup {Q}} \subseteq \overline{\bigcup {F}}=\bigcup _{O\in F}{\overline{O}} \subseteq \bigcup R\), as F is finite. Also \(\bigcup _{O\in F}{\overline{O}}\) is compact, as a finite union of compact sets, and hence \(\bigcup Q\Subset \bigcup R\).
Now each \(T_x\) is nonempty and round by (Cover) and (PointRound). We claim each \(T_x\) is also tight. Indeed, for any finite \(F \subseteq T_x\) and \(H\prec P{\setminus } T_x\), note that \(O\in H\) means \(O\Subset N\not \ni x\), for some \(N\in P\), and hence \(x\notin {\overline{O}}\), as X is Hausdorff. Thus \(x\in \bigcap F{\setminus }\overline{\bigcup H}\) so we have some \(O\in P\) with \(O\Subset \bigcap F{\setminus }\overline{\bigcup H}\), by (Dense), thus verifying (2.1).
Then we immediately see that \(x\mapsto T_x\) is a homeomorphism with respect to the topologies on X and \({\mathcal {T}}(P)\) generated by P and \((O_p)_{p\in P}\) respectively. Thus \(x\mapsto T_x\) is also a homeomorphism with respect to the corresponding patch topologies, which are just the original topologies of X and \({\mathcal {T}}(P)\), by Propositions 2.37 and 2.44. \(\square \)
In summary, a concrete pseudobasis of a locally compact Hausdorff space always yields an abstract pseudobasis. Moreover, if we represent this abstract pseudobasis as a concrete pseudobasis of its tight spectrum, we recover the original pseudobasis.
Remark 2.46
When \(\prec \) is reflexive it suffices to consider \(F=q\) in (Separative), in which case this reduces to the usual notion of a separative poset (see [16, Definition III.4.10]) and \((P,\mathrel {\mathsf {C}})\) is the separative quotient of \((P,\prec )\). In general, if \((P,\prec )\) is an abstract pseudobasis then \((P,\mathrel {\mathsf {C}})\) is again an abstract pseudobasis, which can be seen by going through the representation above or more directly using the basic properties in Proposition 1.5.
4 Inverse semigroups
Next we examine what happens when our abstract pseudobases have some extra inverse semigroup structure. In this case, each tight subset will be a coset which will naturally give the tight spectrum an extra étale groupoid structure. First, however, we need some general results about inverse semigroups and their cosets.
4.1 Cosets
Throughout this section we assume S is an inverse semigroup
Definition 3.1
([21, §3.1]) We call \(C \subseteq S\) a coset if \(CC^{1}C=C^\le =C\).
Proposition 3.2
Proof
Proposition 3.3
If \(T^{1}T \subseteq (CC^{1})^\le \) and C is coset then so is \((TC)^\le \).
Proof
Proposition 3.4
If C is a coset then \(C\cap E\ne \emptyset \) iff \(C=(CC^{1})^\le \).
Proof
As with any inverse semigroup, this restricted product yields a groupoid (see [17, §3.1]), i.e. a set G with an inverse operation and partial product operation making G a small category in which all morphisms are isomorphisms.
Definition 3.5
4.2 Ordered inverse semigroups
We will be interested inverse semigroups with an extra (possibly nonreflexive) order relation \(\prec \) which behaves well with respect to the inverse semigroup structure.
Definition 3.6
Remark 3.7
Any inverse semigroup S becomes an ordered inverse semigroup when we take \(\prec \ =\ \le \). The point is that there may be many other strictly stronger relations \(\prec \) which make S an ordered inverse semigroup.
One immediate consequence is the following auxiliaryproduct property.
Proposition 3.8
Proof
Proposition 3.9
Every Frink filter T in an ordered inverse semigroup is a coset.
Proof
To see that \(T=T^\le \), note that if \(u\ge t\in T\) then, as T is a Frink filter, we have \(w\in P\) and finite \(F \subseteq T\) with \({\widehat{F}}\prec w\prec t\le u\). As \(\prec \) is auxiliary to \(\le \), this yields \({\widehat{F}}\prec w\prec u\) and hence \(u\in T\), as T is a Frink filter.
For the rest of this section we assume that \((S,\prec )\) is an ordered inverse semigroup with zero and \(P=S{\setminus }\{0\}\) is round
Remark 3.10
Proposition 3.11
Proof

(3.3) If \(q,r\ge p\ne 0\) then, as P is round, we have \(p'\ne 0\) with \(q,r\ge p\succ p'\) and hence \(q,r\succ p'\), by (Left Auxiliary), so \(q\not \perp r\). Conversely, if \(q\not \perp r\) then we have \(p\ne 0\) with \(q,r\succ p\) and hence \(q,r\ge p\), as \(\prec \ \subseteq \ \le \). This proves the \(\Leftrightarrow \) part. For \(\Rightarrow \), just note \(q\wedge r=0\) implies \(qs\wedge rs=(q\wedge r)s=0\), by [17, Proposition 1.4.19].
 (3.4) If \(Q\mathrel {\mathsf {D}}R\), i.e. \(Q^\succ {\setminus }\{0\}\succ R^\succ {\setminus }\{0\}\), thenand hence \(Q^\ge {\setminus }\{0\}\ge R^\ge {\setminus }\{0\}\), as P is round and (Left Auxiliary) holds. Conversely, if \(Q^\ge {\setminus }\{0\}\ge R^\ge {\setminus }\{0\}\) then likewise$$\begin{aligned} Q^\ge {\setminus }\{0\}\ \succ \ Q^\succ {\setminus }\{0\}\ \succ \ R^\succ {\setminus }\{0\}\ \subseteq \ R^\ge {\setminus }\{0\} \end{aligned}$$and hence \(Q^\succ {\setminus }\{0\}\succ R^\succ {\setminus }\{0\}\).$$\begin{aligned} Q^\succ {\setminus }\{0\}\ \subseteq \ Q^\ge {\setminus }\{0\}\ \ge \ R^\ge {\setminus }\{0\}\ \succ \ R^\succ {\setminus }\{0\} \end{aligned}$$
Now assume \(Q\mathrel {\mathsf {D}}R\) and \(0\ne p\le qs\), for some \(q\in Q\). Then \(0\ne p=pp^{1}p\le qsp^{1}p\) so \(sp^{1}\ne 0\). Thus \(0\ne ps^{1}\le qss^{1}\le q\). As \(Q\mathrel {\mathsf {D}}R\), we have nonzero \(r\le ps^{1}\) with \(r\le R\). Then again \(0\ne rs\le ps^{1}s\le p\), showing that \(Qs\mathrel {\mathsf {D}}Rs\). \(\square \)
Remark 3.12
Corollary 3.13
Proof

(3.5) If \(Q\mathrel {\mathsf {D}}Q'\) and \(R\mathrel {\mathsf {D}}R'\) then \(QR\mathrel {\mathsf {D}}Q'R\mathrel {\mathsf {D}}Q'R'\), by (3.4) and (1.6) (or rather an extension of (1.6) to infinite unions).

(3.6) If \(Q\mathrel {\mathsf {C}}Q'\) and \(R\mathrel {\mathsf {C}}R'\) then \(Q\mathrel {\mathsf {D}}F\prec Q'\) and \(R\mathrel {\mathsf {D}}G\prec R'\), for some finite \(F,G \subseteq S\), and hence \(QR\mathrel {\mathsf {D}}FG\prec Q'R'\), by (3.5) and (Multiplicative).
Proposition 3.14
Proof
If T is tight then T is a round Frink filter and hence a coset, by Proposition 3.9. Also, for any \(t\in T\), (3.1) yields \(tss^{1}\in Tss^{1} \subseteq T\). As T is round, we have \(u\in T\) with \(u\prec tss^{1}\) and then (3.2) yields \(us\prec tss^{1}s=ts\). This shows that Ts is round and hence so is \((Ts)^\le \), by (Left Auxiliary).
Corollary 3.15
\({\mathcal {T}}(P)\) is a subgroupoid of the coset groupoid of S.
Proof
Take any \(T,U\in {\mathcal {T}}(P)\) with \((T^{1}T)^\le =(UU^{1})^\le \). As U is round, we have \(u\in U\) with \(u\prec U \subseteq S\). Thus \((TU)^\le =(Tu)^\le \in {\mathcal {T}}(P)\), by (3.1) and (3.7). Thus \({\mathcal {T}}(P)\) is closed under the restricted product. By (Invertive), \({\mathcal {T}}(P)\) is also immediately seen to be closed under taking inverses. \(\square \)
Definition 3.16
We call \({\mathcal {T}}(P)\) the tight groupoid of S.
4.3 Étale groupoids
Definition 3.17
Proposition 3.18
G is an étale groupoid if and only if G has an étale subbasis.
Proof
In Hausdorff groupoids even étale pseudosubbases suffice, as we now show.
Proposition 3.19
If G is a coherent étale groupoid then \(G^\mathrm {patch}\) is also étale.
Proof
Next we claim that products of compact saturated subsets are again compact. To see this, take \(C,D\in {\mathcal {C}}(G)\). For each \(c\in C\), we have \(O,N\in \mathcal {OB}(G)\) with \(c\in O\Subset N\), as \(\mathcal {OB}(G)\) covers G and G is locally compact. As C is compact, we can cover it with finitely many such O. This yields a finite \(F \subseteq {\mathcal {C}}(G)\cap \mathcal {OB}(G)^\Supset \) with \(C \subseteq \bigcup F\), i.e. each \(C'\in F\) is compact, saturated and contained in some open bisection. As G is coherent, \(C'\cap C\in {\mathcal {C}}(G)\), for all \(C'\in F\), and \(C=\bigcup _{C'\in F}C'\cap C\). The same applies to D and thus CD can be expressed as a finite union of products of compact saturated subsets, each of which is contained in some open bisection. As \({\mathcal {C}}(G)\) is closed under finite unions, it suffices to consider the case that C and D are already contained in open bisections O and N respectively.
Now consider the maps \(s(g)=g^{1}g\) and \(r(g)=gg^{1}\) restricted to O and N respectively, which are homeomorphisms. In particular, as they are continuous, \(C^{1}C\) and \(DD^{1}\) are compact and also saturated, as mentioned above. As G is coherent, the same applies to their intersection, i.e. \(E=C^{1}C\cap DD^{1}\in {\mathcal {C}}(G)\). Now \(p(g)=s^{1}(g)r^{1}(g)\) restricted to \(O^{1}O\cap NN^{1}\) is also a homeomorphism and hence \(CD=p[E]\) is also compact and saturated, i.e. \(CD\in {\mathcal {C}}(G)\), as required.
Corollary 3.20
If G is Hausdorff with an étale pseudosubbasis P then G is étale.
Proof
By Proposition 3.18, \(G_P(=\)G with the topology generated by P) is étale. By Propositions 2.37 and 3.19, \(G=G_P^\mathrm {patch}\) is étale. \(\square \)
Later we will need the following elementary result on étale families.
Proposition 3.21
Proof
4.4 Pseudobasic inverse semigroups
Now we can finally show that, for suitable inverse semigroups, our tight groupoid is necessarily étale.
Definition 3.22
We call \((S,\prec )\) a pseudobasic inverse semigroup if S is an ordered inverse semigroup with zero and \(P=S{\setminus }\{0\}\) is an abstract pseudobasis.
Theorem 3.23
If S is a pseudobasic inverse semigroup then \({\mathcal {T}}(P)\) is a locally compact Hausdorff étale groupoid with an étale pseudobasis \((O_s)_{s\prec S}\).
Proof
By Proposition 2.17, Corollaries 2.28 and 3.15, \({\mathcal {T}}(P)\) is locally compact Hausdorff groupoid. By Proposition 2.44, \((O_p)_{p\in P}\) and hence \((O_s)_{s\prec S}\) is a pseudo(sub)basis for \({\mathcal {T}}(P)\). In particular, \((O_s)_{s\prec S}\) covers \({\mathcal {T}}(P)\) so, by Proposition 3.20, it suffices to verify that \((O_s)_{s\prec S}\) is an étale family.
Exel’s original tight groupoid construction was obtained from the groupoid of germs coming from the canonical partial action of S on the tight spectrum of E. In the Hausdorff case, this agrees with our tight groupoid when we consider S as a pseudobasic inverse semigroup with \(\prec \ =\ \le \) (see Remarks 2.2 and 3.7). Indeed, it follows from the Hausdorff characterization in [6, Theorem 3.16] that in this case every one of our tight subsets will be a true filter, not just a Frink filter, and these can be identified with germs just like in Lenz’s alternative construction of Paterson’s universal groupoid (see [19] and [21, §3.3]). In the nonHausdorff case, our tight groupoid will instead be a ‘Hausdorffification’ of Exel’s original tight groupoid. To illustrate this, we give a simple example.
Example 3.24
Remark 3.25
If we want a faithful extension of Exel’s original contruction, even in the nonHausdorff case, then this can still be done in a similar way. Basically, the idea is to modify the definition of (both abstract and concrete) pseudobases and work with ‘local’ versions of tight subsets and the tight spectrum, as we discuss in a follow up paper. However, the Hausdorff construction presented here may be of some interest in its own right, even in the original \(\prec \ =\ \le \) case. Indeed, it would be interesting to see if our Hausdorff tight groupoid might be a suitable alternative to some of the nonHausdorff tight groupoids already considered in the literature.
Proposition 3.26
Proof
Example 3.27
Say G is a discrete group acting on some locally compact Hausdorff space X. Given any pseudobasis P for X, we can obtain a larger pseudobasis \(P'\) by taking its closure under finite intersections and images of elements of G (and if X is second countable then we can even take P to be a countable (pseudo)basis which will remain countable when enlarged in this way, as long as G is also countable). Then \(S=(g,O)_{g\in G,O\in P'}\) will form a étale pseudobasis of the transformation groupoid (G, X). Applying Proposition 3.26 allows us to recover (G, X) just from the inverse semigroup structure of S.
Notes
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