General Non-Commutative Locally Compact Locally Hausdorff Stone Duality
aa r X i v : . [ m a t h . L O ] M a r GENERAL NON-COMMUTATIVE LOCALLY COMPACTLOCALLY HAUSDORFF STONE DUALITY
TRISTAN BICE AND CHARLES STARLING
Abstract.
We extend the classical Stone duality between zero dimensionalcompact Hausdorff spaces and Boolean algebras. Specifically, we simultane-ously remove the zero dimensionality restriction and extend to ´etale groupoids,obtaining a duality with an elementary class of inverse semigroups.
Introduction
Motivation.
There has been a recent surge of interest in extending classical Stonedualities between certain lattices and topological spaces to corresponding non-commutative objects like inverse semigroups and ´etale groupoids. This line ofinvestigation started with work of Kellendonk [Kel97] and Lenz [Len08] who aimedto reconstruct tiling groupoids (which are zero-dimensional) from a certain inversesemigroup of its compact bisections defined from the geometry of the tiling in ques-tion. Exel [Exe10] then showed one could always reconstruct a zero-dimensional´etale groupoid from its inverse semigroup of compact bisections. A number ofrecent papers, notably by Kudryavtseva, Lawson, Lenz and Resende, have general-ized these to various settings. For example [Law12] and [KL16] extend the classicalStone duality between generalized Boolean algebras and zero-dimensional locallycompact Hausdorff topological spaces, while [Res07], [LL13] and [KL17] extendthe duality between spatial frames/locales and sober topological spaces. However,even in the classical commutative cases, both these dualities have their drawbacks.Specifically, zero-dimensional spaces are often too restrictive for applications, forexample when we want use ´etale groupoids to define C*-algebras with few projec-tions. On the other hand, while frames describe very general topological spaces,they are often too big, usually uncountable even when the spaces they describe arecountable (e.g. the frame of open sets of Q ). In model theoretic terms the problemis that, while Boolean algebras are first order structures, frames are second orderstructures, relying as they do on infinitary operations, namely infinite joins.Our goal is to show that it is possible to get the best of both worlds, removingthe zero-dimensionality restriction while still obtaining a duality with a first orderstructure. More precisely we show that certain bases of general locally compact Mathematics Subject Classification.
Key words and phrases. compact, Hausdorff, basis, Stone space, rather below, ´etale groupoid,inverse semigroup.The first author is supported by IMPAN (Poland).The second author is supported by a Carleton University internal research grant.This collaboration began at the Fields Institute “Workshop on Dynamical Systems and Oper-ator Algebras” and “Workshop on New Directions in Inverse Semigroups” held at the Universityof Ottawa in May-June 2016. locally Hausdorff ´etale groupoids are dual to a natural first order finitely axioma-tizable class of inverse semigroups. The key is to consider not just the canonicalorder ≤ in the semigroup but also the ‘rather below’ relation ≺ , which allows oneto recover compact containment on the corresponding basis elements.In fact a similar idea, at least in the commutative case, already appears in [Shi52],which has led to the study of ‘compingent algebras’, i.e. Boolean algebras togetherwith an extra proximal neighbourhood relation ≪ - see [Vri62] and [BdR63]. Theseare dual to algebras of regular open sets in compact Hausdorff spaces, where thejoin operation is given by O ∪ N ◦ . In contrast, we consider general open sets wherethe join operation is given simply by the union O ∪ N . We feel this is the morenatural operation to consider, although it would also be interesting to see if asimilar non-commutative extension of Shirota/de Vries duality could be obtainedwith appropriately defined ‘compingent semigroups’. Outline.
In [BS16, § §
4] we obtained a duality between basic lattices and certainbases of locally compact Hausdorff spaces. The first order of business is to extendthis from Hausdorff to locally Hausdorff spaces, which is the content of § § ∨ -semilattices (i.e. only bounded pairs have joins –see [GHK +
03, Definition I-4.5]). To extend the theory to such posets, in § ≺ (see [GHK +
03, Definition I-1-11]), as well as a weaker notion in § § ` of the ‘Hausdorff union’ relation.From § ≺ , whichis the order theoretic analog of ‘compact containment’.The abstract counterparts to the bases we consider are the basic posets we definein §
2. We first investigate a couple of properties specific to basic posets in § § ∪ -bases of locally compact locally Hausdorff spacesare indeed ` -basic posets. It might even be helpful to look at Theorem 2.20 firstto get some idea of the significance of the posets and relations we are considering.To obtain the other direction of the duality, we need to examine filters. The firststep is to show that characterizations of ultrafilters for Boolean algebras extendto basic posets, as shown in Theorem 3.5. With the help of the key Lemma 3.8,we then show in Theorem 3.9 how basic posets indeed become bases of their ≺ -ultrafilter spaces, which are always locally compact locally Hausdorff spaces.This completes duality between ∪ -bases and ` -basic posets, at least as far objectsin the respective categories are concerned. In §
4, we show that this duality isalso functorial for (even partially defined) continuous maps between spaces and
ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 3 certain relational basic morphisms between basic posets. The extension to ‘non-commutative posets’, i.e. inverse semigroups, and ‘non-commutative spaces’, i.e.´etale groupoids, is then dealt with in § § Summary.
The classic Stone duality is usually stated as ‘zero-dimensional locallycompact Hausdorff spaces are dual to generalized Boolean algebras’. But the dualityreally concerns bases rather than the spaces themselves. To make this explicit, letus call a basis of a topological space a clopen ∪ -basis if it consists of compact clopen subsets and is closed under finite unions ∪ . In fact, it is not hard to see thata clopen ∪ -basis must actually consist of all compact clopen subsets. The followingis then a slightly more accurate summary of classic Stone duality. Theorem (Stone) . Clopen ∪ -bases are dual to generalized Boolean algebras. More precisely, every clopen ∪ -basis is a generalized Boolean algebra, when con-sidered as a poset with respect to inclusion ⊆ , and conversely every generalizedBoolean algebra can be represented as a clopen ∪ -basis of some topological space.Of course, to actually have a clopen ( ∪ -)basis the topological space must be zero-dimensional locally compact Hausdorff, in which case the clopen ∪ -basis is unique,which is why such spaces and bases are usually conflated.We consider more general ∪ -bases – see Definition 2.17. Basically, a ∪ -basisconsists of relatively compact Hausdorff subsets and must be closed under finiteunions whenever possible. Thus the spaces that have ∪ -bases are precisely thelocally compact locally Hausdorff spaces – see Proposition 2.19 – the key pointbeing that they no longer need to be zero-dimensional. On the abstract side ofthings, we consider ‘ ` -basic posets’ – see Definition 2.1 – which generalize Booleanalgebras – see § Theorem. ∪ -bases are dual to ` -basic posets.Proof. By Theorem 2.20, every ∪ -basis is a ` -basic poset with respect to the inclu-sion ordering. Conversely, every ` -basic poset can be represented as ∪ -basis of its ≺ -ultrafilter space, by Theorem 3.9. Moreover, if this ≺ -ultrafilter space represen-tation is applied to a ∪ -basis, we recover the original space, by Proposition 3.4. (cid:3) We also have the following non-commutative extension to certain bases of ´etalegroupoids and certain kinds of inverse semigroups.
Theorem. ∪ -´etale bases are dual to ≃ -basic semigroups.Proof. By Theorem 6.12, every ∪ -´etale basis is a ≃ -basic semigroup. Conversely,every ≃ -basic semigroup can be represented as a ∪ -´etale basis of its ≺ -ultrafiltergroupoid, by Theorem 6.22. Moreover, if this ≺ -ultrafilter groupoid representationis applied to a ∪ -basis, we recover the original groupoid, by Proposition 6.18. (cid:3) The classic Stone duality is also functorial with respect to the appropriate mor-phisms, which can be summarized as follows.
Theorem (Stone) . Continuous maps are dual to Boolean homomorphisms. Note for us ‘compact’ just means ‘every open cover has a finite subcover’ or, equivalently,‘every net has a convergent subnet’, i.e. we do not require compact sets to be Hausdorff.
TRISTAN BICE AND CHARLES STARLING
More precisely, any continuous map F : G → H , for zero dimensional compactHausdorff spaces G and H , yields a Boolean homomorphism π : Clopen( H ) → Clopen( G ) defined by π ( O ) = F − [ O ] , and conversely every Boolean homomorphism π : Clopen( H ) → Clopen( G ) arisesin this way from some continuous map F .Here we can not simply replace Clopen( G ) and Clopen( H ) with ∪ -bases S and T as these are not uniquely defined by G and H , so there is no guarantee that F − will take elements of T to elements of S . Instead of homomorphisms, weconsider relational ‘basic morphisms’ ⊏ – see Definition 4.1 – arising from partialcontinuous maps F as follows. O ⊏ N ⇔ O ⊆ F − [ N ] . Theorem.
Partial continuous maps are dual to basic morphisms.Proof.
Every partial continuous map defines a basic ( ∨ -)morphism, by Theorem 4.2.Conversely, every basic morphism defines a partial continuous map on the corre-sponding ≺ -ultrafilter spaces, by Theorem 4.4. (cid:3) Auxiliary Relations
We will deal with quite general posets that are not lattices or even semilattices.However they will still satisfy a version of distributivity with respect to a relation ≺ which is auxiliary to ≤ in the sense of [GHK +
03, Definition I-1-11], namely therather below relation introduced in § From now on assume S is a poset and ≺ ⊆ ≤ is an auxiliary relation, i.e. (Auxiliarity) a ≤ b ≺ c ≤ d ⇒ a ≺ d. In particular, if a ≺ b ≺ c then a ≺ b ≤ c so a ≺ c , i.e. ≺ is transitive. However, ≺ need not be reflexive. Indeed, the only reflexive auxiliary relation is ≤ itself.Incidentally, while posets are usually denoted by letters like P , we use S to em-phasize that we are primarily interested in posets coming from inverse semigroups– see § Distributivity.Definition 1.1.
We say S is ≺ -distributive if, whenever b, c ∈ S have a join b ∨ c ,( ≺ -Distributivity) a ≤ b ∨ c ⇔ ∀ a ′ ≺ a ∃ b ′ ≺ b ∃ c ′ ≺ c ( a ′ ≺ b ′ ∨ c ′ ≺ a ) . At first sight this may seem like a strange notion of distributivity. Indeed, wedo not know if any analog of distributivity for non-reflexive transitive relations hasbeen considered before. However, the reflexive case does reduce to something morefamiliar. Specifically, for ≤ -distributivity we can take a ′ = a to obtain( ≤ -Distributivity) a ≤ b ∨ c ⇔ ∃ b ′ ≤ b ∃ c ′ ≤ c ( a = b ′ ∨ c ′ ) , which is the usual notion of distributivity for ∨ -semilattices. Also, ( ≺ -Distributivity)has several consequences familiar from domain theory. For example, the ⇒ part of( ≺ -Distributivity) in the a = b = c case yields(Interpolation) a ≺ b ⇒ ∃ c ( a ≺ c ≺ b ) . Then the ⇐ part of ( ≺ -Distributivity) can be expressed in a simpler form. ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 5
Proposition 1.2. If (Interpolation) holds then the ⇐ part of ( ≺ -Distributivity) is equivalent to either of the following. ∀ c ≺ a ( c ≤ b ) ⇒ a ≤ b. (Approximation) ∀ c ≺ a ( c ≺ b ) ⇒ a ≤ b. (Lower Order) Proof.
Even without (Interpolation), we have(1.1) (Approximation) ⇒ (Lower Order) ⇒ ( ⇐ in ≺ -Distributivity) . Indeed, the first ⇒ is immediate from ≺ ⊆ ≤ . For the second ⇒ , say (Lower Order)and right side of ( ≺ -Distributivity) hold. So, for all a ′ ≺ a , we have b ′ ≺ b and c ′ ≺ c with a ′ ≺ b ′ ∨ c ′ ≤ b ∨ c and hence a ′ ≺ b ∨ c , by (Auxiliarity). As a ′ wasarbitrary, (Lower Order) yields a ≤ b ∨ c , as required.Lastly, assume (Interpolation) and the ⇐ part of ( ≺ -Distributivity) hold. To seethat (Approximation) then holds, take a and b satisfying ∀ c ≺ a ( c ≤ b ). If a ′ ≺ a then (Interpolation) yields b ′ , b ′′ ∈ S with a ′ ≺ b ′ ≺ b ′′ ≺ a . By assumption, b ′′ ≤ b so a ′ ≺ b ′ ≺ b , by (Auxiliarity). This shows that the right side of ( ≺ -Distributivity)is satisfied for b = c so the ⇐ part yields a ≤ b . (cid:3) The ⇒ part of ( ≺ -Distributivity) for a = b ∨ c also yields(Shrinking) a ≺ b ∨ c ⇒ ∃ b ′ ≺ b ∃ c ′ ≺ c ( a ≤ b ′ ∨ c ′ ) . Conversely, using this we can get ( ≺ -Distributivity) from ( ≤ -Distributivity). Proposition 1.3. If ≺ satisfies (Interpolation) , (Lower Order) and (Shrinking) , ( ≤ -Distributivity) ⇒ ( ≺ -Distributivity) . Proof.
Assume (Interpolation), (Lower Order), (Shrinking) and ( ≤ -Distributivity).The ⇐ part of ( ≺ -Distributivity) is then immediate from Proposition 1.2. For the ⇒ part, say we are given a ′ ≺ a ≤ b ∨ c . By (Interpolation), we have a ′′ ∈ S with a ′ ≺ a ′′ ≺ a . By (Auxiliarity), a ′′ ≺ b ∨ c . By (Shrinking), we have b ′ ≺ b and c ′ ≺ c with a ′′ ≤ b ′ ∨ c ′ . By ( ≤ -Distributivity), we have b ′′ ≤ b ′ and c ′′ ≤ c ′ with a ′′ = b ′′ ∨ c ′′ . Thus (Auxiliarity) yields b ′′ ≺ b, c ′′ ≺ c and a ′ ≺ b ′′ ∨ c ′′ ≺ a, i.e. the right side of ( ≺ -Distributivity) holds, as required. (cid:3) The following result generalizes the fact that, in distributive lattices, not onlydo meets distribute over joins but also joins distribute over meets.
Proposition 1.4. If S is ≺ -distributive and both b ∨ c and b ∨ d exist then a ≤ b ∨ c and a ≤ b ∨ d ⇒ ∀ a ′ ≺ a ∃ e ≺ c, d ( a ′ ≺ b ∨ e ) . Proof.
Given a ′ ≺ a , ( ≺ -Distributivity) applied to a ≤ b ∨ c yields b ′ ≺ b and c ′ ≺ c with a ′ ≺ b ′ ∨ c ′ ≺ a . By ( ≺ -Distributivity) applied to a ′ ≺ b ′ ∨ c ′ ≤ b ′ ∨ c ′ , wehave c ′′ ≺ c ′ with a ′ ≺ b ′ ∨ c ′′ . As c ′′ ≺ c ′ ≤ a ≤ b ∨ d , applying ( ≺ -Distributivity)again yields b ′′ ≺ b and e ≺ d with c ′′ ≺ b ′′ ∨ e ≺ c ′ . By (Auxiliarity), e ≺ c and a ′ ≺ b ∨ e , as a ′ ≺ b ′ ∨ c ′′ ≤ b ′ ∨ b ′′ ∨ e ≤ b ∨ e . (cid:3) TRISTAN BICE AND CHARLES STARLING
Decomposition.Definition 1.5.
We say S has ≺ -decomposition if, whenever b, c ∈ S have a join,( ≺ -Decomposition) a ≤ b ∨ c ⇔ a = _ { a ′ ≺ a : a ′ ≺ b or a ′ ≺ c } . In particular,( ≤ -Decomposition) a ≤ b ∨ c ⇔ a = _ { a ′ ≤ a : a ′ ≤ b or a ′ ≤ c } can be restated as follows whenever a has a meet with both b and c :( ≤ -Decomposition ′ ) a ≤ b ∨ c ⇔ a = ( a ∧ b ) ∨ ( a ∧ c ) . Again, this is the familiar notion of distributivity for lattices.Note that it suffices to verify ⇒ in ( ≺ -Decomposition), as the ⇐ part is imme-diate from ≺ ⊆ ≤ . Likewise, ≺ ⊆ ≤ immediately yields ≥ on the right side so wecan rewrite ( ≺ -Decomposition) equivalently as follows.( ≺ -Decomposition) a ≤ b ∨ c ⇒ a ≤ _ { a ′ ≺ a : a ′ ≺ b or a ′ ≺ c } . Also note that (Approximation) is just ( ≺ -Decomposition) with a = b = c , i.e.(Approximation) ∀ a ∈ S ( a = _ b ≺ a b ) . In fact, ( ≺ -Decomposition) itself follows from ( ≺ -Distributivity). Proposition 1.6.
We have the following general implications. ( ≺ -Distributivity) ⇒ ( ≺ -Decomposition) ⇒ ( ≤ -Decomposition) . Proof.
For the first ⇒ , say a ≤ b ∨ c . For every a ′ ≺ a , ( ≺ -Distributivity) yields b ′ ≺ b and c ′ ≺ c with a ′ ≺ b ′ ∨ c ′ ≺ a . By (Auxiliarity), b ′ , c ′ ≺ a . Also a ′ ≤ b ′ ∨ c ′ ,as ≺ ⊆ ≤ . By Proposition 1.2, ( ≺ -Distributivity) yields (Approximation) so a = _ a ′ ≺ a a ′ ≤ _ { b ′ ∨ c ′ ≺ a : b ′ ≺ b and c ′ ≺ c } ≤ _ { a ′′ ≺ a : a ′′ ≺ b or a ′′ ≺ c } , i.e. ( ≺ -Decomposition) holds. The second ⇒ is immediate from ≺ ⊆ ≤ . (cid:3) Corollary 1.7. If S is a ∧ -semilattice then ( ≺ -Distributivity) ⇒ ( ≤ -Distributivity) ⇔ ( ≤ -Decomposition) . Proof.
The ⇒ follows from Proposition 1.6. The ⇔ follows from the equivalent of( ≤ -Decomposition) given above when a has a meet with b and c . (cid:3) The following is an analog of Proposition 1.4.
Proposition 1.8. If S has ≺ -decomposition and both b ∨ c and b ∨ d exist then a ≤ b ∨ c and a ≤ b ∨ d ⇒ a = _ { a ′ ≺ a : a ′ ≺ b or a ′ ≺ c, d } . Proof.
Assume a ≤ b ∨ c and a ≤ b ∨ d . Take e such that e ≥ a ′ whenever a ′ ≺ a, b or a ′ ≺ a, c, d . We claim that then e ≥ a ′ whenever a ′ ≺ a, c . Indeed, for any a ′ ≺ a, c , we have a ′ ≤ b ∨ d so ( ≺ -Decomposition) yields a ′ = _ { a ′′ ≺ a ′ : a ′′ ≺ b or a ′′ ≺ d } . ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 7 As { a ′′ ≺ a ′ : a ′′ ≺ b or a ′′ ≺ d } ⊆ { a ′′ ≺ a : a ′′ ≺ b or a ′′ ≺ c, d } and e is abovethe latter set, we must have e ≥ a ′ . Thus e ≥ a ′ whenever a ′ ≺ a, b or a ′ ≺ a, c .But a ≤ b ∨ c so by ( ≺ -Decomposition) this implies e ≥ a . As e was arbitrary, a = _ { a ′ ≺ a : a ′ ≺ b or a ′ ≺ c, d } . (cid:3) Hausdorff.
The name for the following is due to (2.6) in Theorem 2.20.
Definition 1.9.
Define the
Hausdorff relation ` on S from ≺ as follows. a ` b ⇔ ∀ a ′ ≺ a ∀ b ′ ≺ b ∃ c ≺ a, b ∀ c ′ ≺ a ′ , b ′ ( c ′ ≺ c ) . In a sense, a ` b is saying that a and b have a meet relative to ≺ rather than ≤ . Proposition 1.10. If ≺ = ≤ then a ` b holds if and only if a ∧ b exists.Proof. If ≺ = ≤ then the right side of the definition of a ` b reduces to the a ′ = a and b ′ = b case, in which case we must have c = a ∧ b by the definition of a meet. (cid:3) Note ≤ is stronger than ` , as witnessed by taking c = a ′ above, i.e.( ≤ ⊆ ` ) a ≤ b ⇒ a ` b. In particular, ` is reflexive. We can also extend (Interpolation) to Hausdorff pairs. Proposition 1.11. (Interpolation) is equivalent to ( ` -Interpolation) a ` b ⇒ ∀ c ≺ a, b ∃ d ( c ≺ d ≺ a, b ) . Proof.
If ( ` -Interpolation) holds then, as ` is reflexive, the a = b case reducesto (Interpolation). Conversely, assume (Interpolation) holds and a ` b . For any c ≺ a, b , (Interpolation) yields a ′ , b ′ ∈ S with c ≺ a ′ ≺ a and c ≺ b ′ ≺ b . Then a ` b yields d ≺ a, b with c ≺ d , as c ≺ a ′ , b ′ . (cid:3) We can then replace the second last ≺ with ≤ in Definition 1.9. Proposition 1.12.
Under (Lower Order) , (Interpolation) is equivalent to ( ` -Equivalent) a ` b ⇔ ∀ a ′ ≺ a ∀ b ′ ≺ b ∃ c ≺ a, b ∀ c ′ ≤ a ′ , b ′ ( c ′ ≺ c ) . Proof. As ` is reflexive, ( ` -Equivalent) with a = b and a ′ = b ′ = c ′ yields(Interpolation).Conversely, assume (Lower Order) and (Interpolation) hold. If a ′ ≺ a ` b ≻ b ′ then we have d ≺ a, b with d ′ ≺ d , for all d ′ ≺ a ′ , b ′ . By ( ` -Interpolation), wehave c with d ≺ c ≺ a, b . Now if c ′ ≤ a ′ , b ′ then, for all d ′ ≺ c ′ , (Auxiliarity)yields d ′ ≺ a ′ , b ′ so d ′ ≺ d . As d ′ was arbitrary, (Lower Order) yields c ′ ≤ d ≺ c so (Auxiliarity) yields c ′ ≺ c . This verifies the ⇒ part of ( ` -Equivalent). The ⇐ part is immediate from ≺ ⊆ ≤ and the defintion of ` in Definition 1.9. (cid:3) When S also has meets, ⇒ becomes ⇔ in ( ` -Interpolation). Put another way, ` is really about ≺ preserving meets. Proposition 1.13. If S is a ∧ -semilattice satisfying (Interpolation) then ( ∧ -Preservation) a ` b ⇔ ∀ a ′ ≺ a ∀ b ′ ≺ b ( a ′ ∧ b ′ ≺ a ∧ b ) . TRISTAN BICE AND CHARLES STARLING
Proof.
Assume S is a ∧ -semilattice satisfying (Interpolation). If a ` b then, forany a ′ ≺ a and b ′ ≺ b , (Interpolation) yields a ′′ , b ′′ ∈ S with a ′ ≺ a ′′ ≺ a and b ′ ≺ b ′′ ≺ b . By (Auxiliarity), a ′ ∧ b ′ ≺ a ′′ , b ′′ so a ` b yields c ≺ a, b with a ′ ∧ b ′ ≺ c . As c ≤ a ∧ b , (Auxiliarity) again yields a ′ ∧ b ′ ≺ a ∧ b .Conversely, if the right side holds then, by (Interpolation), we have some c with a ′ ∧ b ′ ≺ c ≺ a ∧ b , which thus witnesses a ` b , again by (Auxiliarity). (cid:3) Often ` will hold for all bounded pairs (e.g. when ≺ is the rather below relationas in Proposition 1.26 below) meaning that we have(Locally Hausdorff) a, b ≤ c ⇒ a ` b. Then ` will have the appropriate auxiliarity property. Proposition 1.14. If S satisfies (Locally Hausdorff) and (Interpolation) then ( ` -Auxiliarity) a ≤ a ′ ` b ′ ≥ b ⇒ a ` b. Proof.
Say a ′′ ≺ a ≤ a ′ ` b ′ ≥ b ≻ b ′′ . By (Auxiliarity), a ′′ ≺ a ′ ` b ′ ≻ b ′′ so wehave c ′ ≺ a ′ , b ′ with d ≺ c ′ , for all d ≺ a ′′ , b ′′ . Then ( ` -Interpolation) yields c with c ′ ≺ c ≺ a ′ , b ′ . As a, c ≤ a ′ , (Locally Hausdorff) implies a ′′ ≺ a ` c ≻ c ′ , so we have a ′′′′ ≺ a, c with d ≺ a ′′′′ , for all d ≺ a ′′ , c ′ . Again ( ` -Interpolation) yields a ′′′ with a ′′′′ ≺ a ′′′ ≺ a, c . Likewise, as b, c ≤ b ′ , (Locally Hausdorff) implies b ′′ ≺ b ` c ≻ c ′ ,so we have b ′′′′ ≺ b, c with d ≺ b ′′′′ , for all d ≺ b ′′ , c ′ . Yet again ( ` -Interpolation)yields b ′′′ with b ′′′′ ≺ b ′′′ ≺ b, c . As a ′′′ , b ′′′ ≤ c , (Locally Hausdorff) again implies a ′′′′ ≺ a ′′′ ` b ′′′ ≻ b ′′′′ , so we have c ′′′ ≺ a ′′′ , b ′′′ with d ≺ c ′′′ , for all d ≺ a ′′′′ , b ′′′′ .Thus c ′′′ ≺ a ′′′ ≺ a and c ′′′ ≺ b ′′′ ≺ b . Also, any d ≺ a ′′ , b ′′ satisfies d ≺ c ′ andhence d ≺ a ′′′′ , b ′′′′ so d ≺ c ′′′ . Thus c ′′′ witnesses a ` b . (cid:3) Disjoint.
We now wish to examine the disjoint relation ⊥ . The appropriatedefinition of ⊥ depends on whether the poset has a minimum or not. For us,particularly when we deal with inverse semigroups in §
5, it will be more natural toactually have a minimum so from now on assume S has a minimum , i.e. (Minimum) ∀ a ∈ S (0 ≤ a ) . The following concept will play an important role later on.
Definition 1.15.
We call U ⊆ S ≺ -round if( ≺ -Round) ∀ u ∈ U ∃ v ∈ U ( v ≺ u ) . For the moment we simply note that S \ { } is often ≺ -round. Proposition 1.16. If (Approximation) holds then S \ { } is ≺ -round.Proof. Assume (Approximation) holds but S \ { } is not ≺ -round. This meanswe have some non-zero a ∈ S such that b ≺ a only holds possibly for b = 0. By(Approximation), a ≤ a = 0, a contradiction. (cid:3) Definition 1.17.
Define the disjoint relation ⊥ on S by a ⊥ b ⇔ a ∧ b = 0 . ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 9
We immediately see that ⊥ has the following auxiliarity property.( ⊥ -Auxiliarity) a ≤ a ′ ⊥ b ′ ≥ b ⇒ a ⊥ b. As long as 0 ≺ ⊥ is also stronger than ` from Definition 1.9, i.e.( ⊥ ⊆ ` ) a ⊥ b ⇒ a ` b. In contrast to ` , however, ⊥ is rarely reflexive. Also, by Proposition 1.16, if(Approximation) holds then ⊥ can be characterized by ≺ , specifically( ⊥ -Equivalent) a ⊥ b ⇔ ∄ c = 0 ( c ≺ a, b ) . Let us also consider the following properties of ⊥ , whenever b ∨ c exists. a ⊥ b and a ≤ b ∨ c ⇒ a ≤ c. ( ⊥ -Decomposition) a ⊥ b and a ≺ b ∨ c ⇒ a ≺ c. ( ⊥ -Distributivity) a ⊥ b and a ⊥ c ⇒ a ⊥ b ∨ c. ( ∨ -Preservation) Proposition 1.18.
We have the following implications. ( ≤ -Decomposition) ⇒ ( ⊥ -Decomposition) ⇒ ( ∨ -Preservation) . ( ≺ -Distributivity) ⇒ ( ⊥ -Distributivity) . Proof.
If ( ≤ -Decomposition) holds then a ≤ b ∨ c and a ⊥ b implies a = _ { a ′ ≤ a : a ′ ≤ b or a ′ ≤ c } = _ { a ′ ≤ a : a ′ ≤ c } ≤ c, so ( ⊥ -Decomposition) holds. If ( ⊥ -Decomposition) holds then a ⊥ b and a b ∨ c implies we have non-zero a ′ ≤ a with a ′ ≤ a, b ∨ c . Thus a ′ ≤ a ⊥ b so, by( ⊥ -Decomposition), a ′ ≤ c and hence a c , showing that ( ∨ -Preservation) holds.If ( ≺ -Distributivity) holds and a ≺ b ∨ c then (Shrinking) yields b ′ ≺ b and c ′ ≺ c with a ≤ b ′ ∨ c ′ . Then a ⊥ b ≥ b ′ implies a ≤ c ′ ≺ c , by ( ⊥ -Decomposition), so a ≺ c , by (Auxiliarity), showing that ( ⊥ -Distributivity) holds. (cid:3) Rather Below.Definition 1.19.
Define the rather below relation ≺ on S by a ≺ b ⇔ a ≤ b and ∀ c ≥ b ∃ a ′ ⊥ a ( ∃ d ≥ a ′ , b and ∀ d ≥ a ′ , b ( c ≤ d )) . This is a poset analog of ‘compact containment’, as can be seen from (2.5) below.It is immediate from Definition 1.19 that rather below ≺ is auxiliary to ≤ , i.e.(Auxiliarity) a ≤ a ′ ≺ b ′ ≤ b ⇒ a ≺ b. Also note that 0 ≺
0. In the presence of joins, we can simplify the definition of ≺ . Definition 1.20. S is a conditional ∨ -semilattice if bounded pairs have joins, i.e. a, b ≤ c ⇒ a ∨ b exists . When S is a conditional ∨ -semilattice, the rather below relation ≺ is given by(Rather Below) a ≺ b ⇔ a ≤ b and ∀ c ≥ b ∃ a ′ ⊥ a ( c ≤ a ′ ∨ b ) . This form makes it clearer that ≺ is a variant of the rather below relation from[PP12, Ch 5 § ≪ from domain theory, which it is sometimes compared to.For example, on the entire open set lattice of a compact Hausdorff space both therather below and way-below relations coincide with compact containment O ⊆ N . In fact, as we will later note, the rather below relation recovers compact containmenteven on a basis which is just closed under finite unions.However, the same can no longer be said for the way-below relation.
Example 1.21.
Let S be the clopen subsets of the Cantor space C = 2 N . Certainly C ⊆ C , even though C C in S . To see this, pick any point x ∈ C . The elementsof S avoiding x form an ideal I with S I = C \ { x } . As x is not isolated and theelements of S are closed, W I = C in S even though C / ∈ I , i.e. C C .Just as the way-below relation is the strongest possible approximating relation– see [GHK +
03, Proposition I-1.15] – the rather below relation is the strongestpossible ≻ -round ⊥ -distributive relation, at least under appropriate conditions.Note ≻ -round refers to the dual to ( ≺ -Round), i.e. U is ≻ -round if( ≻ -Round) ∀ u ∈ U ∃ v ∈ U ( u ≺ v ) . Proposition 1.22. If S is a conditional ∨ -semilattice, the rather below relation iscontained in all auxiliary ≺ ⊆ ≤ satisfying ( ≻ -Round) and ( ⊥ -Distributivity) . If S is even a ∨ -semilattice satisfying ( ⊥ -Decomposition) then the rather below relationis precisely the intersection of all such ≺ .Proof. Say a is rather below b . As S satisfies ( ≻ -Round), we have c ≻ b . Inparticular, b ≤ c so we must have a ′ ⊥ a with c ≤ a ′ ∨ b . As a ≤ b ≺ c ≤ a ′ ∨ b ,( ⊥ -Distributivity) yields a ≺ b .Now say S is a ∨ -semilattice satisfying ( ⊥ -Decomposition) and a is not ratherbelow b . We need to find auxiliary ≺ ⊆ ≤ with a b satisfying ( ≻ -Round) and( ⊥ -Distributivity). If a (cid:2) b then we can just take ≺ = ≤ . If a ≤ b then we musthave c ≥ b such that there is no a ′ ⊥ a with c ≤ a ′ ∨ b . In this case, define ≺ by d ≺ e ⇔ d ≤ e and ∃ d ′ ⊥ d ( c ≤ d ′ ∨ e ) . Certainly ≺ ⊆ ≤ is auxiliary to ≤ and a b . Also we always have d ≺ c ∨ d (witnessed by d ′ = 0) so ( ≺ -Round) holds. To see that ( ⊥ -Distributivity) holds,say d ⊥ e and d ≺ e ∨ f , so we have d ′ ⊥ d with c ≤ d ′ ∨ e ∨ f . By ( ∨ -Preservation), d ⊥ d ′ ∨ e so it follows that d ≺ f , as required. (cid:3) Let us now stop considering arbitrary auxiliary relations and restrict our atten-tion to the rather below relation.
From now on ≺ specifically refers to the rather below relation. Let us now show that the a ≤ b part of (Rather Below) follows automatically if a and b are bounded, as long as we have some degree of distributivity. Proposition 1.23. If S is a conditional ∨ -semilattice satisfying ( ⊥ -Decomposition) , a ≺ b ⇔ ∃ c ≥ a, b and ∀ c ≥ b ∃ a ′ ⊥ a ( c ≤ a ′ ∨ b ) . Proof.
We immediately have ⇒ . Conversely, assume the right side holds. Then wecan take c ≥ a, b and a ′ ⊥ a with a ≤ c ≤ a ′ ∨ b . Then ( ⊥ -Decomposition) yields a ≤ b and hence a ≺ b , by (Rather Below). (cid:3) Another thing worth pointing out is that in general the inequality c ≤ a ′ ∨ b above can be strict. Although if S were ≤ -distributive then we could replace a ′ with something smaller to obtain c = a ′ ∨ b . Even under ≺ -distributivity, we canstill ensure that a ′ ∨ b is not too large. ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 11
Proposition 1.24. If S is a ≺ -distributive conditional ∨ -semilattice then (Complements) a ≺ b ≤ c ≺ d ⇒ ∃ a ′ ⊥ a ( c ≺ a ′ ∨ b ≺ d ) . Proof.
Assume a ≺ b ≤ c ≺ d . By (Interpolation), we have d ′ with c ≺ d ′ ≺ d .Now take a ′ ⊥ a with d ′ ≤ a ′ ∨ b . As c ≺ d ′ ≤ a ′ ∨ b , ≺ -distributivity yields a ′′ ≺ a ′ and b ′ ≺ b with c ≺ a ′′ ∨ b ′ ≺ d ′ . As b ′ ≺ b ≤ c ≺ d ′ , we have c ≺ a ′′ ∨ b ′ ≤ a ′′ ∨ b ≤ d ′ ≺ d. As a ′′ ≺ a ′ ⊥ a , we also have a ′′ ⊥ a so we are done. (cid:3) Corollary 1.25. If S is a ≺ -distributive conditional ∨ -semilattice then the ratherbelow relation defined within d ≻ = { a ∈ S : a ≺ d } is just the restriction to d ≻ of the rather below relation ≺ defined within S .Proof. From (Complements) it follows that a ≺ b ≺ d implies that a is ratherbelow b in d ≻ . Conversely, the restriction of ≺ to d ≻ is ( ≻ -Round) in d ≻ , by(Interpolation), and also remains ≺ -Distributive and hence ⊥ -Distributive in d ≻ .Thus ≺ must contain the rather below relation in d ≻ , by Proposition 1.22. (cid:3) Note we are now considering ` to be defined specifically from the rather belowrelation too, rather than some arbitrary auxiliary relation. This allows us to provefurther properties of ` . For example, under ( ≺ -Distributivity), ` not only contains ≤ but in fact holds for all bounded pairs. Proposition 1.26. If S is a ≺ -distributive conditional ∨ -semilattice then (Locally Hausdorff) a, b ≤ c ⇒ a ` b. Proof.
Say a ′ ≺ a and b ′ ≺ b , for some a, b ≤ c . By (Rather Below), a ′ ≺ a meanswe have d ⊥ a ′ with c ≤ d ∨ a . As b ′ ≺ b ≤ c ≤ d ∨ a , ( ≺ -Distributivity) yields d ′ ≺ d and a ′′ ≺ a with b ′ ≺ d ′ ∨ a ′′ ≺ b . Note a ′′ ≺ a, b , by (Auxiliarity). Also, forany c ′ ≺ a ′ , b ′ , we have c ′ ≺ a ′ ⊥ d ≻ d ′ and c ′ ≺ b ′ ≺ d ′ ∨ a ′′ and hence c ′ ≺ a ′′ , by( ⊥ -Distributivity). Thus a ′′ witnesses a ` b . (cid:3) If S is even a lattice then ` is trivial, i.e. ≺ preserves meets. Proposition 1.27. If S is a distributive lattice, (Interpolation) implies ` = S × S .Proof. By ( ∧ -Preservation), we have to show that, for all a, b, c ∈ S , a ≺ b, c ⇒ a ≺ b ∧ c. If a ≺ b, c and d ≥ b ∧ c then, as S is a lattice, we can set d ′ = d ∨ b ∨ c . Applying(Rather Below) to a ≺ b and then a ≺ c , we obtain b ′ , c ′ ⊥ a with d ′ ≤ b ′ ∨ b ≤ c ′ ∨ c .By ( ∨ -Preservation), b ′ ∨ c ′ ⊥ a and, by distributivity, d ≤ d ′ ≤ ( b ′ ∨ b ) ∧ ( c ′ ∨ c ) ≤ b ′ ∨ c ′ ∨ ( b ∧ c ) . As d was arbitrary, this shows that a ≺ b ∧ c . (cid:3) A similar argument shows ≺ preserves joins in a distributive lattice. In fact, ≺ -distributivity suffices, so long as S is also ≻ -round. We examine such posets furtherin the next section, but first let us show that ( ≻ -Round) (for S ) is not automaticfrom the other conditions we have considered so far. Example 1.28.
Take a sequence ( a n ) of subsets of N such that a = ∅ and, for all n ∈ N , a n ⊆ a n +1 and a n +1 \ a n is infinite (e.g. a n = N \ { k n − : k ∈ N } ). Let S be the collection of subsets of N with finite symmetric difference with some a n , i.e. S = { b ⊆ N : ( b \ a n ) ∪ ( a n \ b ) is finite, for some n ∈ N . } Considering S as a poset w.r.t. inclusion ⊆ , we immediately see that S is a dis-tributive lattice with minimum ∅ and meets and joins given by ∩ and ∪ . Moreover, ≺ is just the restriction of ⊆ to finite subsets on the left, i.e.(1.2) a ≺ b ⇔ a is a finite subset of b, for all a, b ∈ S . Indeed, if a is finite then, for any c ⊇ a , we have a ′ = c \ a ∈ S so a ′ ⊥ a and c = a ′ ∪ a , i.e. a ≺ a and hence a ≺ b , for any b ⊇ a . On theother hand, if a is infinite then any a ′ ∈ S with a ′ ⊥ a must be finite. Thus, forany b ⊇ a , we can take c ∈ S with c ⊇ b and c \ b infinite, which means there isno a ′ ∈ S with a ′ ⊥ a and c ⊆ a ′ ∪ b . This shows that there is no b ∈ S with a ≺ b , i.e. ( ≻ -Round) fails for infinite a . However, from (1.2) it can also be verifiedthat S satisfies ( ≺ -Distributivity) (note (Lower Order) and hence the ⇐ part of( ≺ -Distributivity) would fail if just considered the poset formed from ( a n ) withoutadding in finite differences). 2. Basic Posets
We are now in a position to examine posets we are primarily interested in.
Definition 2.1.
We call S a ` -basic poset if S is a ≺ -distributive and( ` -Joins) a ∨ b exists ⇔ ∃ a ′ , b ′ ( a ≺ a ′ ` b ′ ≻ b ) , Just to reiterate, ≺ denotes the rather below relation defined from ≤ as inDefinition 1.19 and ` denotes the Hausdorff relation defined from ≺ as in Definition 1.9.Our goal will be to show that ` -basic posets are dual to ‘ ∪ -bases’ (see § ` -basic’. Infact, we will show in Theorem 3.9 that the following slightly more general posetscan also be represented as bases of locally compact locally Hausdorff spaces. Definition 2.2.
We call S a basic poset if(1) S is ≻ -round,(2) S is ≺ -distributive, and(3) S is a conditional ∨ -semilattice. Proposition 2.3.
Every ` -basic poset S is a basic poset.Proof. First note that the ⇒ part of ( ` -Joins), even just in the a = b case, isequivalent to ( ≻ -Round). Thus whenever a, b ≤ c , we have some c ′ ≻ c whichmeans a ≺ c ′ ` c ′ ≻ b and hence a ∨ b exists, by the ⇐ part of ( ` -Joins). Thus S is also a conditional ∨ -semilattice and hence a basic poset. (cid:3) Basic Properties.Proposition 2.4. If S is a basic poset then (Predomain) a, b ≺ c ⇒ a ∨ b ≺ c. ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 13
Proof.
Take a, b ≺ c . As S is ≻ -round, for any d ≥ c , we have e ≻ d . Then wecan take a ′ ⊥ a with e ≤ a ′ ∨ c . As d ≺ e ≤ a ′ ∨ c , (Shrinking) yields a ′′ ≺ a ′ with d ≤ a ′′ ∨ c . Likewise, we can take b ′ ⊥ b with d ≺ e ≤ a ′ ∨ c ≤ b ′ ∨ c = f so(Shrinking) again yields b ′′ ≺ b ′ with d ≤ b ′′ ∨ c . As a ′ , b ′ ≤ f , (Locally Hausdorff)implies a ′′ ≺ a ′ ` b ′ ≻ b ′′ which yields c ′ ≺ a ′ , b ′ with c ′ ≻ c ′′ , for all c ′′ ≤ a ′′ , b ′′ . As c ′ ≺ a ′ ⊥ a and c ′ ≺ b ′ ⊥ b , ( ∨ -Preservation) yields c ′ ⊥ a ∨ b . As d ≤ a ′′ ∨ c, b ′′ ∨ c ,( ≤ -Decomposition) and Proposition 1.8 yields d = _ { c ′′ ≤ d : c ′′ ≤ c or c ′′ ≤ a ′′ , b ′′ } ≤ c ∨ c ′ . As d was arbitrary, a ∨ b ≺ c . (cid:3) In fact, as basic posets satisfy ( ≺ -Distributivity) and hence (Interpolation), a, b ≺ c ⇒ ∃ d ( a, b ≺ d ≺ c ) . This means basic posets are ‘(stratified) predomains’ in the sense of [Kei17] or‘abstract bases’ in the sense of [GL13, Lemma 5.1.32].Generally, to verify a ≺ b we need to consider all c ≥ b . However, for basicposets, it suffices to consider just one c ≻ a . Proposition 2.5. If S is a basic poset then a ≺ b ⇔ ∃ c ≻ a ∃ a ′ ⊥ a ( c ≤ a ′ ∨ b ) . Proof.
The ⇒ part is immediate from ( ≻ -Round). Conversely, if a ′ ⊥ a ≺ c ≤ a ′ ∨ b then ( ⊥ -Distributivity) yields a ≺ b . (cid:3) This is possibly a good point to summarize some of the most important propertieswe have considered so far. Specifically, recall that any basic poset satisfies a ≤ b ≺ c ≤ d ⇒ a ≺ d. (Auxiliarity) ∀ c ≺ a ( c ≤ b ) ⇒ a ≤ b. (Approximation) a ≤ a ′ ` b ′ ≥ b ⇒ a ` b. ( ` -Auxiliarity) c ≺ a, b and a ` b ⇒ ∃ d ( c ≺ d ≺ a, b ) . ( ` -Interpolation) a ≺ b ≤ c ≺ d ⇒ ∃ a ′ ⊥ a ( c ≺ a ′ ∨ b ≺ d ) . (Complements) a, b ≺ c ⇒ a ∨ b ≺ c. (Predomain)2.2. Boolean Examples.
The construction of locally compact locally Hausdorfftopological spaces from basic posets will follow the construction of Stone spacesfrom Boolean algebras. This will come in the next section on filters, although thepotential for doing this is already hinted at by the following.
Proposition 2.6.
Boolean algebras are just bounded basic posets with ≺ = ≤ .Proof. If S is a Boolean algebra then by definition S is a distributive complementedlattice. In particular, S is bounded, i.e. S has a maximum (for complements toeven be defined) and the existence of complements implies that ≺ is ≤ , so S iscertainly ≺ -round, i.e. S is a bounded basic poset.Conversely, if S is a bounded basic poset with ≺ = ≤ then, in particular, S isnot just a conditional ∨ -semilattice but a true ∨ -semilattice. Also ≺ = ≤ implies ≺ is reflexive so every element has a complement and hence S is also a ∧ -semilattice,as a ∧ b = ( a c ∨ b c ) c . Also ≺ -distributivity becomes ≤ -distributivity so S is adistributive complemented lattice, i.e. a Boolean algebra. (cid:3) Proposition 2.7.
Generalized Boolean algebras are just ` -basic posets with ≺ = ≤ and ` = S × S. Proof. If S is a ` -basic poset with ` = S × S then, for any a, b ∈ S , ≻ -roundnessyields a ′ , b ′ ∈ S such that a ≺ a ′ ` b ′ ≻ b . So a ∨ b exists, by ( ` -Joins), showingthat S is a true ∨ -semilattice. If ≺ = ≤ too then ≺ is reflexive so, whenever a ≤ b ,we have a ≺ a ≤ b ≺ b . Then (Complements) yields a ′ ⊥ a with b = a ′ ∨ a , i.e. a ′ = b \ a is a complement of a relative to b . It follows that S is also a ∧ -semilattice,as a ∧ b = a ∨ b \ (( a ∨ b \ a ) ∨ ( a ∨ b \ b )). Thus S is a distributive relativelycomplemented lattice, i.e. a generalized Boolean algebra.Conversely, if S is a generalized Boolean algebra then the existence of relativecomplements yields ≺ = ≤ . Thus S is ≺ -round and ≺ -distributive. Also ≺ = ≤ means that a ` b is just saying a and b have a meet, by Proposition 1.10. As S isa lattice this is always true so ` = S × S . (cid:3) Dropping the ` = S × S condition yields a further ‘locally Hausdorff’ general-ization of Boolean algebras. With this in mind, we make the following definition. Definition 2.8.
We call S locally Boolean or a local Boolean algebra if ↓ a = { b ∈ S : b ≤ a } is a Boolean algebra, for all a ∈ S .In particular, a local Boolean algebra has meets whenever it has joins, i.e. a ∨ b exists ⇒ a ∧ b exists . More interesting is the converse, when S has as many joins as possible. Proposition 2.9.
A local Boolean algebra is just a basic poset with ≺ = ≤ . Inthis case, S is a ` -basic poset if and only if it has joins whenever it has meets, i.e. ( ∧ -Joins) a ∧ b exists ⇒ a ∨ b exists . Proof. If S is locally Boolean then certainly S is a conditional ∨ -semilattice andagain the existence of relative complements yields ≺ = ≤ . Thus S is ≺ -round and ≺ -distributive and hence a basic poset. The converse follows from (Complements)as above. For the last part note again that ≺ = ≤ means that a ` b is just saying a and b have a meet, by Proposition 1.10. So if S is also a ` -basic poset and a ∧ b exists then a ≺ a ` b ≻ b so a ∨ b exists, by ( ` -Joins). Conversely, if joins existwhenever meets do and a ≺ a ′ ` b ′ ≻ b then ( ` -Auxiliarity) yields a ` b , i.e. a ∧ b exists so a ∨ b exists, showing that ( ` -Joins) holds. (cid:3) Example 2.10.
For an example of a local Boolean algebra satisfying ( ∧ -Joins)which is not a generalized Boolean algebra, consider the compact open Hausdorffsubsets S of the ‘bug-eyed’ Cantor space C , i.e. the Cantor space { , } N withan extra copy x ′ of one point x . More precisely, let C be the quotient spaceobtained from two copies { , } N ⊔ { , } N ′ of the Cantor space by identifying each c and c ′ except when c = x . Then C \ { x } and C \ { x ′ } are maximal elements of S , necessarily with no meet (which can also be seen directly from the fact theirintersection C \ { x, x ′ } is not compact).For examples of ` -basic posets where ≺ 6 = ≤ , we turn to ∪ -bases of topologicalspaces. But first we need to examine the compact containment relation. ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 15
Compact Containment.Throughout this section assume G is a topological space . While topological spaces are usually denoted by letters like X , we use G toemphasize that we are primarily interested in spaces coming from ´etale groupoids– see § Definition 2.11.
The compact containment relation ⋐ on subsets of G is given by O ⋐ N ⇔ ∃ compact C ( O ⊆ C ⊆ N ) . We immediately see that compact containment is auxiliary to containment, i.e. O ⊆ O ′ ⋐ N ′ ⊆ N ⇒ O ⋐ N. Compact containment is also related to closures, particularly for Hausdorff subsets.
Proposition 2.12.
For any Hausdorff
O, N, M ⊆ G with O ⊆ N ⋐ M , (2.1) O ⋐ N ⇔ O ∩ N is compact ⇔ O ∩ M ⊆ N. Proof.
Assume O ⋐ N , i.e. O ⊆ C ⊆ N , for some compact C . As N is Hausdorff, C must be relatively closed in N so O ∩ N ⊆ C . As closed subsets of compactspaces are compact, O ∩ N is also compact. Conversely, if O ∩ N is compact then O ⋐ N , as O ⊆ O ∩ N ⊆ N .Again assume O ⋐ N , i.e. we have compact C with O ⊆ C ⊆ N ⊆ M . As M is Hausdorff, C must be relatively closed in M so O ∩ M ⊆ C ⊆ N . Conversely,assume O ∩ M ⊆ N . As N ⋐ M , we have compact C with O ⊆ N ⊆ C ⊆ M .Again C is relatively closed in M , as M is Hausdorff, so O ∩ N = O ∩ M ⊆ C iscompact, as a closed subset of C . (cid:3) We call G locally compact if every point g ∈ G has a neighbourhood base ofcompact subsets. In other words, G is locally compact if each neighbourhood filter N g = { N ⊆ G : g ∈ N ◦ } is ⋐ -round . Remark 2.13.
In Hausdorff spaces it is common to use weaker notions of localcompactness like in Proposition 2.19 (3) below. However, the stronger notion givenhere is standard for more general potentially non-Hausdorff spaces – see [GHK + Proposition 2.14. If G is locally compact and T with basis S , TFAE. (1) G is Hausdorff. (2) Every compact subset of G is closed. (3) For every
O, N ∈ S , O ⋐ N ⇒ O ⊆ N. Proof. (1) ⇒ (2) Standard, even without local compactness.(2) ⇒ (3) If O ⊆ C ⊆ N , for compact C , then (2) implies C is closed so O ⊆ C ⊆ N .(3) ⇒ (1) Say (1) fails, i.e. G is not Hausdorff, so we have g, h ∈ G whose neighbour-hoods are never disjoint. As G is T , we can assume w.l.o.g. that we havesome N ∈ S with g / ∈ N ∋ h . As G is locally compact, we have O ∈ S with h ∈ O ⋐ N . By assumption, every neighbourhood of g intersects O so g ∈ O \ N , i.e. O N so (3) fails. (cid:3) Now we want to relate ⋐ with ≺ on suitable bases S of G , taking inclusion ⊆ asthe order relation ≤ which in turn defines the rather below relation ≺ . We can noteimmediately that arbitrary bases will not do, e.g. any maximal O ∈ S will triviallysatisfy O ≺ O , even when O is not compact, i.e. O ⋐ O . We avoid this problem bydealing with ⋑ -round bases that are closed under bounded finite unions. Proposition 2.15. If S is a basis of G that is closed under bounded finite unions(i.e. O ∪ N ∈ S whenever O, N ⊆ M and O, N, M ∈ S ) then, for any O, N, M ∈ S , (2.2) O ≺ N ⋐ M ⇒ O ⊆ N ⋐ M and O ∩ M ⊆ N ⇒ O ⋐ N. If G is also ⋑ -round then, for any O, N ∈ S , (2.3) O ≺ N ⇔ O ⊆ N and [ N ⋐ M ∈ S O ∩ M ⊆ N ⇒ O ⋐ N. Proof.
Assume
O, N, M ∈ S and O ≺ N ⋐ M . Then certainly O ≤ N , i.e. O ⊆ N ,and we must have compact C with N ⊆ C ⊆ M as well as L ∈ S with O ∩ L = ∅ and M ⊆ N ∪ L (as S is closed under bounded finite unions, ∨ is ∪ ). Thus O ∩ L = ∅ and hence O ⊆ O ∩ C ⊆ O ∩ M ⊆ N . Also, as a closed subset of a compact space, O ∩ C is compact and hence O ⋐ N , thus proving (2.2).Now assume that G is also ⋑ -round. The ⇒ parts of (2.2) follow immediatelyfrom (2.3). Conversely, for the first ⇐ in (2.2), assume O ⊆ N and(2.4) [ N ⋐ M ∈ S O ∩ M ⊆ N. Given M ∈ S with N ⊆ M , we can take P ∈ S with M ⋐ P , as S is ⋑ -round, so wehave some compact Hausdorff C with M ⊆ C ⊆ P . By (2.4), O ∩ C ⊆ O ∩ P ⊆ N so C \ N is disjoint from O . As C \ N is relatively closed in C it is also compact.Thus, for each g ∈ C \ N ⊆ P , we have some O g ∈ S with g ∈ O g ⊆ P \ O . So the( O g ) cover C \ N and we must have some finite subcover O g , . . . , O g n . As M ⊆ C ,they also cover M \ N , and as they are all contained in P , their union L must bein S . As L covers M \ N , we have M ⊆ N ∪ L as well as O ∩ L = ∅ . As M wasarbitrary, this shows that O ≺ N . (cid:3) Example 2.16.
To see that ⋐ can be strictly weaker than ≺ above, we can againconsider the bug-eyed Cantor space C from Example 2.10, this time taking S to bethe collection of all (even non-Hausdorff) compact open subsets. Thus S is closedunder arbitrary finite unions and is also certainly ⋑ -round, as O ⋐ O ⋐ C ∈ S , forall O ∈ S . In particular, O ⋐ O for O = C \ { x ′ } ∈ S , even though O ∩ C = C O and hence O O .2.4. ∪ -Bases.Throughout this section again assume G is a topological space . Definition 2.17.
We call a basis S ∋ ∅ of G a ∪ -basis if, for all O, N ∈ S ,( ∪ ) O ∪ N ∈ S ⇔ O ∪ N ⊆ C, for some compact Hausdorff C ⊆ G. In particular, taking O = N in the ⇒ part of ( ∪ ), we see that every element ofa ∪ -basis is required to be contained in some compact Hausdorff subset. Then the ⇐ part of ( ∪ ) ensures that S is a conditional ∨ -semilattice.For Hausdorff G , ∪ -bases are just bases which consist of relatively compact opensubsets and which are closed under taking finite unions. Note this union condition ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 17 is crucial if we hope to recover the space from the order structure of the basis, asarbitrary bases may fail to distinguish spaces like the unit interval from the Cantorspace. Indeed, any second countable compact Hausdorff space has a countable basisof regular open sets. This generates a countable Boolean algebra of regular opensets, which is atomless as long as the space has no isolated points. However, allcountable atomless Boolean algebras are isomorphic.We call G locally Hausdorff if every point has a Hausdorff neighbourhood. Notepoints are, in particular, compact Hausdorff. In locally Hausdorff spaces, it turnsout that all compact Hausdorff subsets have Hausdorff neighbourhoods. In partic-ular, we can always extend C in ( ∪ ) to some open Hausdorff subset to obtain( ∪ ) O ∪ N ∈ S ⇔ O ∪ N ⋐ M, for some open Hausdorff M ⊆ G. Proposition 2.18. If G is locally Hausdorff then every compact Hausdorff subset C ⊆ G is contained in some open Hausdorff subset.Proof. For every g ∈ C , take some Hausdorff open O g ∋ g . As C is Hausdorff, forevery h ∈ C \ O g , we have disjoint open N h ∋ g and M h ∋ h . As C is compact and O g is open, C \ O g is compact so we have some finite subcover M ′ g = M h ∪ . . . ∪ M h k of C \ O g . Let N ′ g = O g ∩ N h ∩ . . . ∩ N h k . Again as C is compact, we have somefinite subcover N = N ′ g ∪ . . . ∪ N ′ g j of C . Let O = N ∩ ( O g ∪ M ′ g ) ∩ . . . ∩ ( O g j ∩ M ′ g j ) . Certainly O is an open subset containing C . To see that O is Hausdorff, take any g, h ∈ O . As g ∈ N , we have g ∈ N ′ g l for some l . We also have h ∈ O g l ∪ M ′ g l . If h ∈ O g l then, as g ∈ N ′ g l ⊆ O g l too and O g l is Hausdorff, we can separate h and g with disjoint open sets. If h ∈ M ′ g l then, as N ′ g l ∩ M ′ g l = ∅ , we already have disjointopen subsets separating h and g . As g and h were arbitrary, we are done. (cid:3) Proposition 2.19.
The following are equivalent. (1) G has a ∪ -basis. (2) G is locally compact locally Hausdorff. (3) Every point g ∈ G has a compact Hausdorff neighbourhood.Proof. (1) ⇒ (3) Immediate.(3) ⇒ (2) Take g ∈ G with a compact Hausdorff neighbourhood C and an openneighbourhood O . As C is Hausdorff, for each c ∈ C \ O , we have disjointopen O c ∋ g and N c ∋ c . As C \ O is closed in C and C is compact, C \ O is also compact so we have some finite subcover N c , . . . , N c k of C \ O . Let O ′ = O c ∩ . . . ∩ O c k . As C is compact, O ′ ∩ C is compact.Also O ′ ∩ ( N c ∪ . . . ∪ N c k ) = ∅ so g ∈ O ′ ∩ C ⊆ O . As O was arbitrary,this shows that each g has a neighbourhood base of compact subsets, asrequired.(2) ⇒ (1) If G is locally compact locally Hausdorff then the collection of all opensubsets of G contained in some compact Hausdorff C forms a ` -basis. (cid:3) Note (3) ⇒ (2) is a minor extension of a standard argument for Hausdorff spaces.Also note that (2) ⇔ (3) is essentially sayinglocally compact locally Hausdorff ⇔ locally (compact Hausdorff) . It is also worth noting that in the non-Hausdorff settting, local compactness doesnot follow from compactness. In fact, there is even an example in [Sco13] of alocally Hausdorff space that is compact but not locally compact.
Theorem 2.20.
Let G be a topological space, and let S be a ∪ -basis S of G . Then S is a ` -basic poset (taking ⊆ for ≤ ) such that, for any O, N ∈ S , O ≺ N ⇔ O ⋐ N. (2.5) O ` N ⇔ O ∪ N is Hausdorff . (2.6) Proof.
First we note that S is ⋑ -round. Indeed, by the equivalent version of ( ∪ )above, any O ∈ S must be compactly contained in some open Hausdorff M , i.e. O ⊆ C ⊆ M , for some compact C . As S is a basis, we can cover C with elementsof S compactly contained in M . As C is compact, we can find a finite subcoverwhose union F is compactly contained in M . Thus O ⋐ F ∈ S , by ( ∪ ).Now we show that S is a ` -basic poset satisfying (2.5) and (2.6).(2.5) Take O, N ∈ S . If O ≺ N then O ⋐ N , by (2.3). Conversely, if O ⋐ N then, for every M ∈ S with N ⋐ M , we have O ∩ M ⊆ N , by (2.1). Thus S N ⋐ M ∈ S O ∩ M ⊆ N so O ≺ N , by the other direction of (2.3).( ≻ -Round) As we already know S is ⋑ -round, this is immediate from (2.5).( ≺ -Distributivity) As S is a basis and G is locally compact, every O ∈ S is the union of thoseelements of S compactly contained within O . Thus (Approximation) holds,which yields the ⇐ part of ( ≺ -Distributivity), by (1.1).Conversely, take M, N, O, O ′ ∈ S with O ′ ⋐ O ⊆ N ∪ M ∈ S , so wehave compact C with O ′ ⊆ C ⊆ O . As S is a basis, we can cover C withelements of S which are compactly contained within O and either N or M .As C is compact, we have a finite subcover. As S is a ∪ -basis, we have F ⋐ N , G ⋐ M and F, G ∈ S for the unions of those elements in the finitesubcover compactly contained within N and M respectively. Moreover, O ′ ⋐ F ∪ G ⋐ N ∪ M (and F, G, F ∪ G ∈ S , as N ∪ M ∈ S and S is a ∪ -basis). As M , N , O and O ′ were arbitrary, ( ≺ -Distributivity) holds.(2.6) To prove (2.6), take O, N ∈ S and assume O ` N but O ∪ N is notHausdorff. As O and N are Hausdorff, we must have g ∈ O \ N and h ∈ N \ O with no disjoint neighbourhoods. Take O ′ , N ′ ∈ S with g ∈ O ′ ⋐ O and h ∈ N ′ ⋐ N so O ′ ∩ N ′ ⋐ O, N . For every open M ∋ g , M ∩ O ′ is aneighbourhood of g so M ∩ O ′ ∩ N ′ = ∅ , by the choice of g and h . Thus g ∈ O ′ ∩ N ′ . As O ` N holds, we have M ∈ S such that P ⋐ M ⋐ O, N , for all P ∈ S with P ⋐ O ′ , N ′ . As O ′ ∩ N ′ is the union of such P , we must have O ′ ∩ N ′ ⊆ M and hence g ∈ M . By ( ` -Interpolation), we also have M ′ ∈ S with M ⋐ M ′ ⋐ O, N . Thus g ∈ M ∩ O ⊆ M ′ , as M ⋐ M ′ ⋐ O (see (2.1)),even though g / ∈ N ⊇ M ′ , a contradiction.Conversely, take O, N ∈ S and assume O ∪ N is Hausdorff. Take O ′ , N ′ ∈ S with O ′ ⋐ O and N ′ ⋐ N . In particular, O ∩ N ⋐ O ∪ N so C = O ′ ∩ N ′ ∩ ( O ∪ N )is compact (see (2.1)). Moreover, as O ′ ∩ N ′ ⋐ O, N and O ∪ N is aHausdorff extension of both O and N , we have C ⊆ O ∩ N (see the proofof (2.1)). Thus we can cover C with elements of S compactly contained ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 19 in O ∩ N . As C is compact, we can take a finite subcover whose union F is in S , as S is a ∪ -basis. Then O ′ ∩ N ′ ⋐ F ⋐ O, N and hence, for all M ∈ S with M ⋐ O ′ , N ′ , we have M ⋐ F . As O ′ and N ′ were arbitrary,this shows that O ` N holds.( ` -Joins) Take O, O ′ , N, N ′ ∈ S with O ⋐ O ′ ` N ′ ⋑ N . By what we just proved, O ′ ∪ N ′ is Hausdorff. As O ∪ N ⋐ O ′ ∪ N ′ and S is a ∪ -basis, we must have O ∪ N ∈ S , as required. (cid:3) Incidentally, to get a ∪ -basis S of some compact locally Hausdorff G , it mightbe tempting to simply take the collection of all open Hausdorff subsets. However,unless G itself is Hausdorff, this may fail to yield a ∪ -basis or a ` -basic poset and ⋐ may no longer correspond to ≺ on S as above. Example 2.21.
Consider two copies of [ − ,
1] and identify the strictly negativepoints to form a topological space G = [ − , ⊔ [0 , ′ . It is easy to see that G iscompact and locally Hausdorff but not Hausdorff, as the two copies 0 and 0 ′ of theorigin can not be separated by open sets. Let S be the basis of G consisting of allopen Hausdorff subsets. Then O = G \ { ′ } ∈ S even though the only compactsubset containing O is G , which is not Hausdorff. Thus S is not a ∪ -basis. Moreover, O is a maximal element of S and hence, trivially, O ≺ O in S , even though O isnot compact. Also O ≺ O = [ − , ∪ (0 , ′ even though any N ≺ (0 , ′ wouldhave to avoid a neighbourhood of 0 ′ (as (0 , ′ ⊆ [ − , ∪ [0 , ′ ∈ S so there needsto be some N ⊥ with N ⊥ ∩ N = ∅ and [ − , ∪ [0 , ′ ⊆ N ⊥ ∪ (0 , ′ , which implies0 ′ ∈ N ⊥ ). In particular, we can not have both N ≺ (0 , ′ and O ⊆ [ − , ∪ N , i.e. S fails to satisfy ( ≺ -Distributivity) or even (Shrinking).3. Filters
Definition 3.1.
For any transitive relation < on S , we call U ⊆ S a < -filter if v > u ∈ U ⇒ v ∈ U ( > -Closed) u, v ∈ U ⇒ ∃ w ∈ U ( w < u, v ) . ( < -Directed)Note we can extend any U ⊆ S to a > -closed subset by taking the union with U < = { a > u : u ∈ U } . Also denote the initial segment of U defined by any a ∈ S by U a = U ∩ a > = { u ∈ U : u < a } . Initial segments will play an important role in § Proposition 3.2.
For any U ⊆ S and any transitive relation < on S , U is a < -filter ⇔ ∀ a ∈ U ( U = U a< ) . Proof.
Say U is a < -filter and a ∈ U . Then U is < -directed so, for every u ∈ U , wehave v ∈ U with v < a, u which means v ∈ U a and hence u ∈ U a< , i.e. U ⊆ U a< .On the other hand, U a ⊆ U and hence U a< ⊆ U , as U is > -closed, i.e. U = U a< .Conversely, assume U satisfies the right side. For any a, b ∈ U , we have b ∈ U a< so there must be some u ∈ U a with u < b . But this means u ∈ U and u < a , so U is < -directed. As U = U a< , U is also > -closed and hence a < -filter. (cid:3) Ultrafilters.Definition 3.3. A < -ultrafilter is a maximal proper < -filter.Ultrafilters can be used to recover the points of a space from a basis, as shownby the following generalization of [BS16, Proposition 3.2]. Proposition 3.4. If S ∋ ∅ is a Hausdorff basis of some locally compact G then g U g = { O ∈ S : g ∈ O } is a bijection from G to ⋐ -ultrafilters of S .Proof. If U is a ⋐ -ultrafilter of S then take M ∈ U . By Proposition 3.2 above, for U M = U ∩ M ⋑ = { O ∈ U : O ⋐ M } , we have \ U = \ U M ⊆ \ O ∈ U M O ∩ M = \ O ∈ U M O ⋐ N ∈ U O O ∩ M ⊆ \ N ∈ U M N = \ U M , where the last ⊆ follows form (2.1). As an intersection of non-empty compact closedsubsets of O with the finite intersection property, T U must be non-empty. For any g ∈ T U , we have U ⊆ U g and hence, by maximality, U = U g .Conversely, as G is locally compact, each U g is a ⋐ -filter and hence has somemaximal extension U ′ ⊇ U g not containing ∅ , by the Kuratowski-Zorn lemma. Bywhat we just proved, U ′ = U g ′ for some g ′ ∈ G . But as G is locally Hausdorff andhence T , U g ⊆ U g ′ implies g = g ′ . Thus U g was already a ⋐ -ultrafilter. (cid:3) Now take a poset S . As the rather below relation ≺ is auxiliary to ≤ , U is a ≺ -filter ⇔ U is a ≺ -round ≤ -filter . As 0 ≺
0, the proper ≺ -filters are precisely those that do not contain 0. Thus every ≺ -filter not containing 0 is contained in a ≺ -ultrafilter, again by the Kuratowski-Zorn lemma. In fact, the same applies to ≺ -directed subsets, as the upwards ≺ -closure of a ≺ -directed subset is a ≺ -filter.It is well-known that proper filters in Boolean algebras are maximal (i.e. ultrafil-ters) iff they are prime iff they intersect every complementary pair. This generalizesto basic posets and even ≺ -distributive conditional ∨ -semilattices as follows. Theorem 3.5. If U is a proper ≺ -filter in a ≺ -distributive conditional ∨ -semilattice S then U being maximal is equivalent to each of the following conditions. a ≺ b / ∈ U ⇒ ∃ c ∈ U ( a ⊥ c ) or ∀ c ∈ U ( b ` c ) . (Complementary) a ∨ b ∈ U ⇒ a ∈ U or b ∈ U. (Prime) Proof.
Assume that (Complementary) holds but U is not maximal, so we have aproper ≺ -filter V properly containing U . Take v ∈ V \ U and u ∈ U . Then we have w ∈ V with w ≺ v, u . In particular, w ≺ v / ∈ U . Having c ∈ U ⊆ V with w ⊥ c would contradict the fact V is ≺ -filter, so (Complementary) implies that w ` c , forall c ∈ U . In particular, w ` u , even though w ≤ u , contradicting ( ≤ ⊆ ` ). Thus(Complementary) implies maximality.Conversely, assume that U is a ≺ -filter failing (Complementary), so we have a, b, c ∈ S with a ≺ b / ∈ U , b ` c ∈ U and a u , for all u ∈ U . To see that U is notmaximal, let V = { v ∈ S : ∃ d ( a ≺ d ≺ b and ∃ u ∈ U ∀ e ≺ d, u ( e ≺ v )) } . ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 21
We immediately see that V is ≻ -closed. Also 0 / ∈ V , otherwise we would have d with a ≺ d ≺ b and u ∈ U such that e ≺ d, u is only possible for e = 0. By( ⊥ -Equivalent), this would mean d ⊥ u and hence a ⊥ u , a contradiction. Next weclaim that V is ≺ -directed. To see this, take v, v ′ ∈ V and corresponding d, d ′ ∈ S with a ≺ d, d ′ ≺ b and u, u ′ ∈ U with e ≺ v , for all e ≺ d, u , and e ≺ v ′ , for all e ≺ d ′ , u ′ . As d, d ′ ≺ b , (Locally Hausdorff) from Proposition 1.26 yields d ` d ′ and then ( ` -Interpolation) yields d ′′ , d ′′′ ∈ S with a ≺ d ′′′ ≺ d ′′ ≺ d, d ′ . As U isa ≺ -filter, we also have u ′′′ , u ′′ ∈ U with u ′′′ ≺ u ′′ ≺ u, u ′ , c . As d ′′ ≺ b ` c ≻ u ′′ ,( ` -Auxiliarity) implies d ′′′ ≺ d ′′ ` u ′′ ≻ u ′′′ , so we have v ′′ ≺ d ′′ , u ′′ with e ≺ v ′′ ,for all e ≺ d ′′′ , u ′′′ . Thus v ′′ ≺ d ′′ ≺ d ′ , d and v ′′ ≺ u ′′ ≺ u ′ , u so v ′′ ≺ v ′ , v . Also d ′′′ and u ′′′ witness v ′′ ∈ V so the claim is proved and we have shown that V is aproper ≺ -filter. Moreover, U ⊆ V and b ∈ V \ U , so V is a proper extension of U ,i.e. U is not maximal. In other words, maximality implies (Complementary).(Prime) ⇒ (Complementary) Assume (Prime), a ≺ b / ∈ U and b ` u ∈ U . We have to find v ∈ U with a ⊥ v . Take u ′ ∈ U with u ′ ≺ u . As a ≺ b ` u ≻ u ′ , we have c ′ ≺ b, u with e ≺ c ′ , for all e ≺ a, u ′ . By ( ` -Interpolation), we have c with c ′ ≺ c ≺ b, u .As c ′ ≺ c ≤ u , (Rather Below) yields d ⊥ c ′ such that c ∨ d ≥ u ∈ U . As c ≤ b / ∈ U , we must have d ∈ U , by (Prime). As U is a ≺ -filter, we have v ∈ U with v ≺ d, u ′ . Then, for any e ≺ a, v , we have e ≺ a, u ′ and hence e ≺ c ′ ⊥ d ≻ v ≻ e so e = 0. By ( ⊥ -Equivalent), a ⊥ v ∈ U as required.(Complementary) ⇒ (Prime) Assume (Complementary) and a ∨ b ∈ U but a, b / ∈ U , looking for a con-tradiction. Take c ∈ U with c ≺ a ∨ b . By (Shrinking), we have a ′ ≺ a and b ′ ≺ b with c ≤ a ′ ∨ b ′ and hence a ′ ∨ b ′ ∈ U . By ( ≤ ⊆ ` ), a ′ ` a ∨ b ∈ U so, as a ′ ≺ a / ∈ U , (Complementary) yields v ∈ U with a ′ ⊥ v . Likewise, wehave w ∈ U with b ′ ⊥ w . As U is a ≺ -filter, we have x ∈ U with x ≺ v, w and hence x ⊥ a ′ ∨ b ′ , by ( ∨ -Preservation), contradicting a ′ ∨ b ′ ∈ U . (cid:3) Ultrafilter Spaces.Let G be the set of ≺ -ultrafilters of a basic poset S . We consider G with the topology generated by ( O a ) a ∈ S where O a = { U ∈ G : a ∈ U } . Note this topology makes the map g U g in Proposition 3.4 a homeomorphism(as the map then takes sets in the given basis to the subbasis ( O a ) a ∈ S ). Definition 3.6.
We call G the ≺ -ultrafilter space of S .Note that when S is a Boolean algebra, G is just the Stone space of S . Our goalis to show that ≺ -ultrafilter space G is still locally compact and locally Hausdorfffor general basic poset S (although, in contrast to Stone spaces, G may not be0-dimensional). First we note some basic facts about G . Proposition 3.7.
The sets ( O a ) a ∈ S form a basis for G and, for all a, b ∈ S , O a ∨ b = O a ∪ O b . (3.1) a ⊥ b ⇔ O a ∩ O b = ∅ . (3.2) Proof. If U ∈ O a ∩ O b then a, b ∈ U so, as U is a ≺ -filter, we have c ∈ U with c ≺ a, b and hence U ∈ O c ⊆ O a ∩ O b . Thus ( O a ) a ∈ S form a basis.(3.1) Immediate from (Prime). (3.2) If a ⊥ b then certainly O a ∩ O b = ∅ . Conversely, if a b then ( ⊥ -Equivalent)and Proposition 1.16 yields a ≺ -decreasing sequence a, b ≻ c ≻ c ≻ . . . ofnon-zero elements, which extends to a ≺ -ultrafilter in O a ∩ O b . (cid:3) Before proving our main theorem we need the following result.
Lemma 3.8.
Assume a ≺ b and C ⊆ S . If a W F , for any finite F ⊆ C , then O a ∩ O b [ c ∈ C O c . Proof.
First note that we can replace C with C ′ = C ≻ ∩ b ≻ = { d ≺ b, c : c ∈ C } .Indeed, if we had finite F ′ ⊆ C ′ with a ≺ W F ′ then we would have finite F ⊆ C with F ′ ⊆ F ≻ and hence a ≺ W F , a contradiction. Moreover, O b ∩ [ c ∈ C O c = O b ∩ [ c ′ ∈ C ′ O c ′ , as any ≺ -filter containing b and c ∈ C must also contain some c ′ ≺ b, c , and hence O a ∩ O b \ [ c ∈ C O c = O a ∩ O b \ [ c ′ ∈ C ′ O c ′ . In particular, if the right side is non-empty then so is the left side.Thus we may assume C is actually bounded by b , i.e. C ⊆ b ≻ . Now consider D = { d ≤ b : F ⊆ C is finite and a ≺ d ∨ _ F } . First we see that 0 / ∈ D , as a W F , for any finite F ⊆ C . Next we claimthat D is ≺ -directed. To see this, take x, y ∈ D , so we have finite F, G ⊆ C with a ≺ x ∨ W F, y ∨ W G ≤ b . By (Locally Hausdorff) from Proposition 1.26, x ∨ W F ` y ∨ W G so ( ` -Interpolation) yields z with a ≺ z ≺ x ∨ W F, y ∨ W G .By Proposition 1.4, we have z ′ ≺ x, y with a ≺ z ′ ∨ W ( F ∪ G ) ≤ b . This provesthe claim, so D can be extended to a ≺ -ultrafilter U . As b ∈ D ⊆ U (e.g. taking F = ∅ ), we have U ∈ O b .Looking for a contradiction, assume that U / ∈ O a . Thus we have u ∈ U with O a ∩ O u = ∅ and hence a ⊥ u , by (3.2). As b ∈ U , by lowering u if necessary, wemay assume that u ≤ b . Take w ∈ U with w ≺ u . By (Rather Below), we have x ⊥ w with b ≤ x ∨ u . As a ≺ b , ( ≺ -Distributivity) yields x ′ ≺ x and u ′ ≺ u with a ≺ x ′ ∨ u ′ ≺ b . As a ⊥ u ≻ u ′ , ( ⊥ -Distributivity) implies a ≺ x ′ and hence x ′ ∈ D ⊆ U (witnessed by F = ∅ ). But this contradicts x ′ ≺ x ⊥ w ∈ U .Lastly, take any x ≺ c ∈ C . By (Rather Below), we have d ⊥ x with a ≺ b ≤ c ∨ d so ( ≺ -Distributivity) yields c ′ ≺ c and d ′ ≺ d with a ≺ c ′ ∨ d ′ ≺ b . Thus a ≺ c ∨ d ′ ≤ b so d ′ ∈ D ⊆ U and hence d ∈ U , which in turn yields x / ∈ U . As x was arbitrary, c / ∈ U , i.e. U / ∈ O c . Putting this all together, we see that U ∈ O a ∩ O b \ [ c ∈ C O c . (cid:3) Theorem 3.9.
Let S be a basic poset and let G be the set of ≺ -ultrafilters of S .Then G is a locally compact locally Hausdorff space and a ≤ b ⇔ O a ⊆ O b . (3.3) a ` b ⇔ O a ∪ O b is Hausdorff . (3.4) a ≺ b ⇔ O a ⋐ O b . (3.5) ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 23
Moreover, if S is a ` -basic poset then ( O a ) a ∈ S is a ∪ -basis for G .Proof. (3.3) If a ≤ b then certainly O a ⊆ O b . Conversely, if a (cid:2) b then we have a ′ ≺ a with a ′ b , by (Lower Order). By Lemma 3.8, with a ′ , a and b taking theplace a , b and C respectively, O a ′ ∩ O a * O b and hence O a * O b . ⇒ in (3.4) Assume a ` b and take any distinct U ∈ O a and V ∈ O b . By maximality, V * U so we can take v ∈ V \ U . Further take v ′ , v ′′ ∈ V with v ′′ ≺ v ′ ≺ v, b so v ′′ ≺ v ′ / ∈ U and v ′ ≺ b ` a . By ( ` -Auxiliarity), v ′ ` a ∈ U so(Complementary) yields c ∈ U with v ′′ ⊥ c . Thus O c and O v ′′ are disjointneighbourhoods of U and V respectively. As a ` a and b ` b , the sameargument would apply to distinct filters both in O a or O b .(3.5) Take a ≺ b . First note that, for any c ≤ b or even c ` b ,(3.6) a ≺ c ⇒ O a ∩ O b ⊆ O c . Indeed, U ∈ O a means that, for every d ∈ U , O d ∩ O a = ∅ so d a , by(3.2). If we also had U ∈ O b \ O c , i.e. a ≺ c / ∈ U but c ` b ∈ U , then(Complementary) would yield d ∈ U with d ⊥ a , a contradiction.Now we want to show that O a ∩ O b is compact and hence O a ⋐ O b . Forthis, we need to show that if ( O c ) c ∈ C has no finite subcover of O a ∩ O b thenthe entirety of ( O c ) c ∈ C does not cover O a ∩ O b either. In fact it suffices toconsider C ⊆ b ≥ , as ( O c ) c ≤ b forms a basis for the subspace topology on O b (as each U ∈ O b is ≺ -directed). But then having no finite subcover impliesthat a W F , for any finite F ⊆ C . Indeed, if we had a ≺ W F , for somefinite F ⊆ C ⊆ b ≥ , then (3.1) and (3.6) would yield O a ∩ O b ⊆ O W F = [ f ∈ F O f , a contradiction. Thus the entirety of ( O c ) c ∈ C does not cover O a ∩ O b either,by Lemma 3.8, showing that O a ∩ O b is indeed compact.Conversely, say a b . If a (cid:2) b then (3.3) yields O a * O b and socertainly O a ⋐ O b . If a ≤ b then, by ( ≻ -Round), we can take c ≻ b , whichthen implies a ≺ c and hence O a ⋐ O c . By Lemma 3.8, with c and b takingthe place of b and C respectively, O a ∩ O c * O b . As c ` c , the ⇒ partof (3.4) implies that O c is Hausdorff so compact subsets are closed. Thus O a ∩ O c is the smallest compact subset of O c containing O a so O a ⋐ O b . ⇐ in (3.4) Now we know that ≺ corresponds to ⋐ , we can argue as in the proof of(2.6) from Theorem 2.20.By (3.5) G is locally compact. Indeed, if U ∈ O a then we have b ∈ U with b ≺ a so U ∈ O b ⋐ O a . Likewise, by (3.4) and the reflexivity of ` , G is locally Hausdorff.Now assume S is a ` -basic poset. To see that ( O a ) a ∈ S is a ∪ -basis, we need toverify ( ∪ ). For the ⇒ part, say O a ∪ O b ⊆ O c , for some a, b, c ∈ S . By ( ≻ -Round),we have d ≻ c and hence O a ∪ O b ⊆ O c ⋐ O d , as required.Conversely, to verify the ⇐ part of ( ∪ ), take a, b ∈ S such that O a ∪ O b ⋐ M ,for some open Hausdorff M . So we have compact C with O a ∪ O b ⊆ C ⊆ M . Foreach U ∈ C we have c ∈ U with O c ⊆ M . Then we can take c ′ ∈ U with c ′ ≺ c . As C is compact, we can cover C with finitely many such subsets O c ′ .Say C is covered by just pair of subsets O c ′ and O d ′ . Then c ′ ≺ c ` d ≻ d ′ ,as O c ∪ O d ⊆ M is Hausdorff, and hence c ′ ∨ d ′ exists in S , by ( ` -Joins). As O a , O b ⊆ C ⊆ O c ′ ∨ d ′ , we have a, b ≤ c ′ ∨ d ′ and hence a ∨ b exists too, as S is aconditional ∨ -semilattice.Now say C is covered by 3 such subsets, O c ′ , O c ′ and O c ′ . By (Interpolation),we have x , x ∈ S with c ′ ≺ x ≺ c and c ′ ≺ x ≺ c . Again by ( ` -Joins), c ′ ∨ c ′ and x ∨ x exist and, by Proposition 2.4, c ′ ∨ c ′ ≺ x ∨ x . As O x ∨ x ∪ O c isHausdorff, still being contained in M , we can apply ( ` -Joins) again to show that c ′ ∨ c ′ ∨ c ′ exists in S . Again O a , O b ⊆ C ⊆ O c ′ ∨ c ′ ∨ c ′ shows that a, b ≤ c ′ ∨ c ′ ∨ c ′ so a ∨ b exists, as S is a conditional ∨ -semilattice. Extending by induction showsthat, regardless of how many subsets are needed to cover C , O a ∪ O b = O a ∨ b isalways in the basis ( O a ) a ∈ S , as required. (cid:3) Functoriality
Up till now we have focused on certain structures rather than the maps betweenthem. However, the classic Stone duality is also functorial with respect to theappropriate morphisms. Specifically, any continuous map F : G → H , for zerodimensional compact Hausdorff spaces G and H , yields a Boolean homomorphism π : Clopen( H ) → Clopen( G ) defined by π ( O ) = F − [ O ] . However, this does not extend to general bases S and T of G and H for thesimple reason that F − may not take elements of T to elements of S . To avoid thisproblem, we instead consider the relation ⊏ between S and T defined by O ⊏ N ⇔ O ⊆ F − [ N ] . Fortunately, such relations admit a first order characterisation.4.1.
Basic Morphisms.Definition 4.1.
For posets S and S ′ , we call ⊏ ⊆ S × S ′ a basic morphism if a ⊏ ⇒ a = 0 . (Faithful) a ≤ b ⊏ b ′ ≤ a ′ ⇒ a ⊏ a ′ . (Auxiliarity) a ≺ b ⊏ c ′ , d ′ ⇒ ∃ b ′ ( a ⊏ b ′ ≺ c ′ , d ′ ) . (Pushforward) a ≺ b ⊏ c ′ ∨ d ′ ⇒ ∃ c ⊏ c ′ ∃ d ⊏ d ′ ( a ≺ c ∨ d ) . ( ∨ -Pullback)We call ⊏ a basic ∨ -morphism if, moreover, whenever the given joins exist, a ⊏ a ′ and b ⊏ b ′ ⇒ a ∨ b ⊏ a ′ ∨ b ′ . ( ∨ -Preserving) ∀ b ≺ a ( b ⊏ a ′ ) ⇒ a ⊏ a ′ . (Lower Relation) a ′ ∈ S ′ ⇒ ⊏ a ′ . (Bottom)As in the previous section, for any U ⊆ S , let U ⊏ = { a ′ ⊐ a : a ∈ U } . Theorem 4.2.
Let G and G ′ be topological spaces, and suppose S and S ′ are ∪ -bases of G and G ′ respectively. Then any continuous φ : D → G ′ , for D ⊆ G ,defines a basic ∨ -morphism ⊏ ⊆ S × S ′ by (4.1) a ⊏ a ′ ⇔ a ⊆ φ − [ a ′ ] . ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 25
Moreover, for every ≺ -ultrafilter U ⊆ S , \ U ∈ D ◦ ⇔ U ⊏ = ∅ , (4.2) in which case φ ( \ U ) = \ U ⊏ . (4.3) Proof.
First we verify the required properties of ⊏ .(Faithful) If a ⊆ φ − [ ∅ ] = ∅ then a = ∅ .(Auxiliarity) If a ≤ b ⊏ b ′ ≤ a ′ then a ⊆ b ⊆ φ − [ b ′ ] ⊆ φ − [ a ′ ] and hence a ⊆ φ − [ a ′ ].(Pushforward) If a ⋐ b ⊆ φ − [ c ′ ] ∩ φ − [ d ′ ] then we have compact C ⊆ G with a ⊆ C ⊆ b .As φ is continuous, φ [ C ] is also compact and φ [ C ] ⊆ φ [ b ] ⊆ c ′ ∩ d ′ . Thuswe can cover φ [ C ] by a finite collection of elements in the basis S ′ whichare compactly contained in c ′ ∩ d ′ . Taking their union, we obtain b ′ ∈ S ′ with φ [ a ] ⊆ φ [ C ] ⊆ b ′ ⋐ c ′ ∩ d ′ and hence a ⊏ b ′ ≺ c ′ , d ′ .( ∨ -Pullback) If a ⋐ b ⊆ φ − [ c ′ ∪ d ′ ] = φ − [ c ′ ] ∪ φ − [ d ′ ] then we can argue as in the proofof ( ≺ -Distributivity) in Theorem 2.20 to obtain c ⋐ φ − [ c ′ ] and d ⋐ φ − [ d ′ ]within b with a ⋐ c ∪ d and hence a ≺ c ∨ d .( ∨ -Preservation) If a ⊆ φ − [ a ′ ] and b ⊆ φ − [ b ′ ] then a ∪ b ⊆ φ − [ a ′ ] ∪ φ − [ b ′ ] = φ − [ a ′ ∪ b ′ ].(Lower Relation) If b ⊆ φ − [ a ′ ], for all b ⋐ a , then a = S b ⋐ a b ⊆ φ − [ a ′ ].(4.2) and (4.3) If T U / ∈ D ◦ then no u ∈ U is contained in D and so we can not have u ⊆ φ − [ a ′ ]( ⊆ D ) for any a ′ ∈ S ′ , i.e. U ⊏ = ∅ . While if g ∈ T U ∈ D ◦ then φ − [ a ′ ] is a neighbourhood of g whenever φ ( g ) ∈ a ′ ∈ S ′ . ByProposition 3.4, U = U g = all those neighbourhoods of g in S so we musthave some u ∈ U with g ∈ u ⊆ φ − [ a ′ ]. Thus U ⊏ = ∅ and φ ( T U ) = φ ( g ) = T U ⊏ , as a ′ was arbitrary and G ′ is locally Hausdorff and hence T . (cid:3) (Bottom) Just note ∅ ⊆ φ − [ a ′ ], for any a ′ ∈ S ′ .Any relation ⊏ ⊆ S × S ′ between posets S and S ′ has an extension ⊑ defined by a ⊑ a ′ ⇔ ∀ b ≺ a ∃ finite F ⊏ a ′ ( b ≺ _ F ) , where F ⊏ a ′ means a ⊏ a ′ , for all a ∈ F , i.e. F ⊆ a ′ ⊐ . Proposition 4.3. If ⊏ is a basic ∨ -morphism between basic posets S and S ′ then ⊑ = ⊏ . Proof.
Taking a for F shows that ⊏ ⊆ ⊑ . Now assume ⊏ is a basic ∨ -morphismand a ⊑ a ′ so, for every b ≺ a , we have finite F ⊏ a ′ with b ≺ W F . If F = ∅ then b ≺ W ∅ = 0 so b = 0 ⊏ a ′ , by (Bottom). If F = ∅ then W F ⊏ a ′ , by ( ∨ -Preserving),so again b ⊏ a ′ , by (Auxiliarity). Thus a ⊏ a ′ , by (Lower Relation). (cid:3) In fact, even when ⊏ is only a basic morphism between basic posets, ⊑ will alwaysbe a basic ∨ -morphism extension. To prove this, we could work directly with theposets S and S ′ . Alternatively, one can go through the ≺ -ultrafilter spaces andnote that this follows from Theorem 4.2 above and (4.6) below. Theorem 4.4.
Let S and S ′ be basic posets, and let G and G ′ be their ≺ -ultrafilterspaces. Then any basic morphism ⊏ defines a continuous map φ from the opensubset D = { U ∈ G : U ⊏ = ∅} to G ′ by (4.4) φ ( U ) = U ⊏ . We can characterize when φ is defined everywhere (i.e. D = G ) as follows. (4.5) ∀ U ∈ G ( U ⊏ = ∅ ) ⇔ ∀ a ∈ S ∃ finite F ⊆ S ′ ⊐ ( a ≺ _ F ) . Moreover, for any a ∈ S and a ′ ∈ S ′ , (4.6) a ⊑ a ′ ⇔ O a ⊆ φ − [ O a ′ ] . Proof.
First note that U ⊏ = ∅ means a ⊏ a ′ , for some a ∈ U and a ′ ∈ S ′ . Thus U ⊏ = ∅ for any U ∈ O a , so D is indeed open.Next we show that U ⊏ ∈ G ′ , for any U ∈ D . By (Faithful), 0 / ∈ U ⊏ , as 0 / ∈ U .If a ′ , b ′ ∈ U ⊏ then we have a, b ∈ U with a ⊏ a ′ and b ⊏ b ′ . Then we have c, d ∈ U with c ≺ d ≺ a, b . By (Auxiliarity), c ≺ d ⊏ a ′ , b ′ so we have d ′ with c ⊏ d ′ ≺ a ′ , b ′ , by (Pushforward), and hence d ′ ∈ U ⊏ . Thus U ⊏ is ≺ -directed andalso ≻ -closed, again by (Auxiliarity). Lastly, if a ′ ∨ b ′ ∈ U ⊏ then we have c, d ∈ U with c ≺ d ⊏ a ′ ∨ b ′ . By ( ∨ -Pullback), we have a ⊏ a ′ and b ⊏ b ′ with c ≺ a ∨ b .By (Prime), either a ∈ U or b ∈ U and hence a ′ ∈ U ⊏ or b ′ ∈ U ⊏ . Thus U ⊏ alsosatisfies (Prime) and hence U ⊏ is a ≺ -ultrafilter, i.e U ⊏ ∈ G ′ .To see that φ is continuous note that, for any a ′ ∈ S ′ , φ − [ O a ′ ] = [ a ⊏ a ′ O a . In particular, φ − [ O a ′ ] is always open.(4.5) If the right side of (4.5) holds then, whenever a ∈ U ∈ G , we can take finite F ⊆ S ′ ⊐ with a ≺ W F . By (Prime), we have some b ∈ F ∩ U and thenwe have some a ′ ∈ S ′ with b ⊏ a ′ . Thus a ′ ∈ U ⊏ = ∅ . Conversely, if theright side of (4.5) fails then we have some a ∈ S such that a W F , for allfinite F ⊆ S ′ ⊐ . By ( ≻ -Round), we have some b ≻ a . By Lemma 3.8 with C = S ′ ⊐ , we have some U ∈ O a ∩ O b \ [ c ∈ C O c ⊆ G \ [ c ⊏ a ′ ∈ S ′ O c . Thus U ⊏ = ∅ .(4.6) If a ⊑ a ′ and U ∈ O a then we have b ∈ U with b ≺ a . Thus we have finite F ⊏ a ′ with b ≺ W F and hence we have some c ∈ F ∩ U . This means a ′ ∈ c ⊏ ⊆ U ⊏ = φ ( U ), i.e. φ ( U ) ∈ O a ′ so U ∈ φ − [ O a ′ ]. Conversely, if a a ′ then we have some b ≺ a such that b W F , for any finite F ⊏ a ′ .By Lemma 3.8 with a and b switched and C = a ′ ⊐ , O a ⊇ O b ∩ O a * [ c ∈ C O c = [ c ⊏ a ′ O c = φ − [ O a ′ ] . Thus O a * φ − [ O a ′ ]. (cid:3) Composition.Definition 4.5.
For any relations ⊏ ⊆ S × S ′ and ⊏ ′ ⊆ S ′ × S ′′ , define a ( ⊏ ◦ ⊏ ′ ) a ′′ ⇔ ∃ a ′ ∈ S ′ ( a ⊏ a ′ ⊏ ′ a ′′ ) . Note this relation ⊏ ◦ ⊏ ′ ⊆ S × S ′′ is the standard composition of the relations ⊏ and ⊏ ′ , which extends the usual notion of function composition. Proposition 4.6. If ⊏ ⊆ S × S ′ and ⊏ ′ ⊆ S ′ × S ′′ are basic morphisms betweenbasic posets S , S ′ and S ′′ then their composition ⊏ ◦ ⊏ ′ is also a basic morphism.Proof. We verify the required properties of ⊏ ◦ ⊏ ′ as follows.(Faithful) If a ⊏ a ′ ⊏ ′ a ′ = 0 and hence a = 0, by (Faithful) for ⊏ and ⊏ ′ , so(Faithful) also holds for ⊏ ◦ ⊏ ′ . ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 27 (Auxiliarity) If a ≤ b ⊏ b ′ ⊏ ′ b ′′ ≤ a ′′ then a ⊏ b ′ ⊏ ′ a ′′ , by (Auxiliarity) for ⊏ and ⊏ ′ ,so (Auxiliarity) also holds for ⊏ ◦ ⊏ ′ .(Pushforward) If a ≺ b ⊏ b ′ ⊏ ′ c ′′ , d ′′ then (Pushforward) for ⊏ yields a ′ with a ⊏ a ′ ≺ b ′ ⊏ ′ c ′′ , d ′′ . Then (Pushforward) for ⊏ ′ yields b ′′ with a ⊏ a ′ ⊏ b ′′ ≺ c ′′ , d ′′ ,so (Pushforward) also holds for ⊏ ◦ ⊏ ′ .( ∨ -Pullback) Assume a ≺ b ⊏ c ′ ⊏ ′ d ′′ ∨ e ′′ . By (Interpolation), we can take f with a ≺ f ≺ b ⊏ c ′ . By (Pushforward) for ⊏ , we have b ′ with f ⊏ b ′ ≺ c ′ ⊏ d ′′ ∨ e ′′ .By ( ∨ -Pullback) for ⊏ ′ , we have d ′ ⊏ ′ d ′′ and e ′ ⊏ ′ e ′′ with f ⊏ b ′ ≺ d ′ ∨ e ′ .By (Auxiliarity) for ⊏ , a ≺ f ⊏ d ′ ∨ e ′ . By ( ∨ -Pullback) for ⊏ , we have d ⊏ d ′ and e ⊏ e ′ with a ≺ d ∨ e . In other words, given a ≺ b ⊏ ◦ ⊏ ′ d ′′ ∨ e ′′ ,we have found d ⊏ ◦ ⊏ ′ d ′′ and e ⊏ ◦ ⊏ ′ e ′′ with a ≺ d ∨ e , as required. (cid:3) Unfortunately, this does not extend to basic ∨ -morphisms, as ( ∨ -Preserving) and(Lower Relation) are not always preserved under composition. Example 4.7.
To see that this can fail for (Lower Relation), let G = G ′ = G ′′ bethe Cantor space C , let S = S ′′ be the ` -basic poset of all open subsets, let S ′ bethe Boolean algebra of just the clopen subsets, let ⊏ and ⊏ ′ be the restrictions ofinclusion ⊆ to S × S ′ and S ′ × S ′′ , and take any non-closed open O . For every open N ⋐ O , we can find clopen O ′ with N ⊆ O ′ ⊆ O , even though there is no singleclopen O ′ with O ⊆ O ′ ⊆ O . Thus ⊏ ◦ ⊏ ′ fails to satisfy (Lower Relation). Example 4.8.
For a (necessarily non-Hausdorff) example of a composition failing( ∨ -Preserving), consider the ‘bug-eyed interval’ G ′ = { ∗ } ∪ [0 ,
1] where 0 ∗ is anextra copy of 0, i.e. { ∗ } ∪ (0 ,
1] is homoemorphic to [0 ,
1] via the map taking 0 ∗ to 0. Let φ be the continuous map from G = [0 ,
1] to G ′ which maps [0 , ] to [0 , ,
1] to { ∗ } ∪ (0 ,
1] by going back and forth one time in the obvious way. Inparticular, φ (0) = 0 and φ (1) = 0 ∗ . Let φ ′ be the countinuous map from G ′ to G ′′ = [0 ,
1] which is the identity on [0 , ∗ to 0. Let S , S ′ and S ′′ be the corresponding ` -basic posets of allHausdorff open sets and define basic ∨ -morphisms ⊏ and ⊏ ′ form φ and φ ′ as in(4.1). Then[0 , ) ⊏ [0 , ⊏ ′ [0 ,
1] and ( , ⊏ { ∗ } ∪ (0 , ⊏ ′ [0 , , even though there is no Hausdorff open O ′ ⊆ G ′ with [0 , ⊏ O ′ ⊏ ′ [0 , ⊏ ◦ ⊏ ′ fails to satisfy ( ∨ -Preserving).This leaves us with two options:(1) Just accept that the same continuous map could arise from different basicmorphisms. Given that we already have to accept that the same topologycan arise from different bases, this is perhaps not a serious drawback.(2) Take ⊏ ◦ ⊏ ′ as the product in the category of ` -basic posets. The drawbackhere is that the definition of ⊑ from ⊏ is not elementary, as it involvesarbitrarily large finite subsets (although it is still first order in the weaksense of being defined by ‘omitting types’ – see [Mar02] Ch 4).In any case, we should still note that composition of basic morphisms correspondsto composition of the resulting continuous maps. By (4.4), this amounts to U ⊏ ◦ ⊏ ′ = ( U ⊏ ) ⊏ ′ , for all ≺ -ultrafilters U ⊆ S , which is immediate from the definitions. Inverse Semigroups
So far we have exhibited a duality between certain posets and topological spaces.Now we wish to obtain a non-commutative extension to certain inverse semigroupsand topological groupoids. So let us now assume S is an inverse semigroup. We denote the idempotents of S by E .We always consider S as a poset with respect to the canonical ordering given by a ≤ b ⇔ a ∈ Eb.
This allows us to apply all the order theory developed so far. Indeed, most of thework has already been done, the only thing left to do is investigate how our previousduality interacts with the multiplicative and inverse structure of S .We take [Law98] as our standard reference for inverse semigroups. A standardexample is the symmetric inverse monoid I ( X ) of all partial bijections on someset X (indeed every inverse semigroup can be represented as a subsemigroup of asymmetric inverse monoid, by the Wagner-Preston theorem – see [Law98, Theorem1.5.1]). As far as the order structure is concerned, I ( X ) is a local Boolean algebraand ∧ -semilattice, so ≺ = ≤ and ` = I ( X ) × I ( X ), by Proposition 1.10.5.1. Distributivity.Definition 5.1.
We call an inverse semigroup S distributive if the product preservesexisting joins, i.e. if a ∨ b exists then so does ac ∨ bc and ca ∨ cb and(Distributivity) ( a ∨ b ) c = ac ∨ bc and c ( a ∨ b ) = ca ∨ cb. Actually, one-sided distributivity suffices, as a a − is an automorphism from S to S op . Also, taking c = ( a ∨ b ) − in (Distributivity) yields(5.1) ( a ∨ b ) − ( a ∨ b ) = ( a − a ) ∨ ( b − b ) , which actually holds even without distributivity – see [Law98, § § E are not always joins in S . Example 5.2.
Let S = { , , e , e , e , s } be the inverse semigroup of partial bijec-tions on X = { , , , , } where 0 is the empty function, 1 is the identity, e k is theidentity restricted to { k } , and s is the total bijection leaving 1, 2 and 3 fixed butswitching 4 and 5. Then e ∨ e = 1 in E but not in S , as e , e ≤ s (cid:2)
1. Indeed,there are no non-trival joins in S (i.e. a ∨ b exists iff a ≤ b or b ≤ a ) so S is triviallydistributive, even though E = { , , e , e , e } is the diamond lattice, which is notdistributive.Thus the (1) ⇒ (2) part of [Law98, § S is a conditional ∨ -semilattice. Proposition 5.3. If S is a conditional ∨ -semilattice, the following are equivalent. (1) E is distributive. (2) S is distributive. Strictly speaking, ‘extension’ may not be the right word, as we did not require our posets tohave meets, and only the ∧ -semilattices can be considered as (commutative) inverse semigroups. ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 29 (3) S satisfies ( ≤ -Distributivity) . (4) S satisfies ( ≤ -Decomposition) .Proof. (1) ⇒ (2) Take a, b ∈ S with a join a ∨ b (in S ). For any c ∈ S , we immediately have ac, bc ≤ ( a ∨ b ) c . On the other hand, for any d ≥ ac, bc ,( a ∨ b ) − dc − ≥ ( a ∨ b ) − acc − = a − acc − . ( a ∨ b ) − dc − ≥ ( a ∨ b ) − bcc − = b − bcc − . So by (5.1) and (Distributivity) in E ,( a ∨ b ) − ( a ∨ b ) cc − = ( a − a ∨ b − b ) cc − = a − acc − ∨ b − bcc − ≤ ( a ∨ b ) − dc − . Multiplying on the left by ( a ∨ b ) and on the right by c we obtain( a ∨ b ) c ≤ ( a ∨ b )( a ∨ b ) − dc − c ≤ d. As d was arbitrary, ( a ∨ b ) c is a join of ac and bc .(2) ⇒ (3) Say a ≤ b ∨ c . By distributivity, a = aa − a = aa − b ∨ aa − c . As aa − b = a ∧ b and aa − c = a ∧ c , by [Law98, § ⇒ (4) See Proposition 1.6.(4) ⇒ (1) Note it suffices to show that E satisfies ( ≤ -Decomposition ′ ), as E is a ∧ -semilattice in which products are meets. As S satisfies ( ≤ -Decomposition),for this it suffices to show that any join in E is a join in S . But if e ∨ f = g in E then, in particular, e and f are bounded so we have some a ∈ S with e ∨ f = a in S (note this is where we need the conditional ∨ -semilatticeassumption). By (5.1), a ∈ E and hence g ≤ a . Conversely, as E ⊆ S , a ≤ g and hence a = g , as required. (cid:3) Rather Below.
Now assume S is an inverse semigroup with zero . By definition, this means we have 0 ∈ S with 0 a = 0, for all a ∈ S . Thus 0 isthe minimum of S in the canonical ordering. In particular, we can again definethe rather below relation ≺ as in Definition 1.19. Although, as before, we willusually assume S is a conditional ∨ -semilattice and then use the equivalent formin (Rather Below).In inverse semigroups, ≺ is determined, at least partially, by its restriction to E . Proposition 5.4. If S is a distributive conditional ∨ -semilattice and a ≤ b then a − a ≺ b − b ⇒ a ≺ b. Proof.
Assume a ≤ b and a − a ≺ b − b . If c ≥ b then c − c ≥ b − b so we have a ′ ⊥ a − a with c − c ≤ a ′ ∨ b − b . Thus ca ′ ⊥ ca − a ≥ ba − a ≥ aa − a = a and c = cc − c ≤ ca ′ ∨ cb − b = ca ′ ∨ b . As c was arbitrary, a ≺ b . (cid:3) However, the reverse implication is not automatic – see Example 6.13. For themoment, let us define a stronger relation ≺≺ by a ≺≺ b ⇔ a − a ≺ b − b and a ≤ b. Basic Semigroups.Definition 5.5.
The compatibility relation ∼ is defined on inverse semigroups by a ∼ b ⇔ ab − , a − b ∈ E. Note ∼ is analogous to ` , e.g. ≤ ⊆ ∼ and, by [Law98, § ∼ -Auxiliarity) a ≤ a ′ ∼ b ′ ≥ b ⇔ a ∼ b. Let us denote the combination of ∼ and ` by ≃ = ∼ ∩ ` , i.e. a ≃ b ⇔ a ∼ b and a ` b. Definition 5.6.
We call S a ≃ -basic semigroup if S is ≺ -distributive and( ≃ -Joins) a ∨ b exists ⇔ ∃ a ′ , b ′ ( a ≺≺ a ′ ≃ b ′ ≻≻ b ) . The motivation for ( ≃ -Joins) is that we are imagining S to consist of openHausdorff bisections that are closed under finite unions ‘whenever possible’, i.e.whenever the union itself is compactly contained in some open Hausdorff bisection– see Definition 6.5 (3 ′ ) below.Again, for the most part, we actually work with slightly more general semigroups. Definition 5.7.
We call S a basic semigroup if(1) S is ≻≻ -round,(2) S is ≺ -distributive, and(3) S is a conditional ∨ -semilattice. Proposition 5.8.
Every ≃ -basic semigroup is a basic semigroup.Proof. See the proof of Proposition 2.3. (cid:3)
In basic semigroups we can forget about the distinction between ≺ and ≺≺ . Proposition 5.9. If S is a basic semigroup then S is a basic poset and (Bi-Below) a ≺ b ⇔ a ≺≺ b. Proof.
Assume S is a basic semigroup. In particular, as S is ≻≻ -round, S is also ≻ -round, by Proposition 5.4. Thus S is basic poset.If a ≺ b then we have c ≥ b with c − c ≻ b − b , as S is ≻≻ -round. As a ≺ b ≤ c ,we have d ⊥ a with c ≤ b ∨ d and hence a − a ≤ b − b ≺ c − c ≤ ( b ∨ d ) − ( b ∨ d ) = b − b ∨ d − d. As a, d ≤ b ∨ d and d ⊥ a , we have d − d = ( b ∨ d ) − d ⊥ ( b ∨ d ) − a = a − a . Thusas a − a ≺ b − b ∨ d − d , ( ⊥ -Distributivity) yields a − a ≺ b − b so a ≺≺ b . (cid:3) It also does not matter if we consider ≺ defined within S or within E . Proposition 5.10. If S is a basic semigroup then E is a basic poset and e ≺ f in E ⇔ e ≺ f in S. Proof.
Take e, f ∈ E with e ≺ f (in S ). For any g ∈ E with g ≥ f , we have h ∈ E with h ≻ g , as S is ≻≻ -round. Then (Complements) yields e ′ ⊥ e with g ≺ e ′ ∨ f ≺ h . Thus f ∈ E , as f ≤ h ∈ E . As g was arbitrary, this shows e ≺ f also holds in E . ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 31
Thus E is ≻ -round, as S is ≻≻ -round. Also, E is a conditional ∨ -semilattice as S is. In fact, by (5.1), joins in S of elements of E must also lie in E . This means( ≺ -Distributivity) in S implies ( ≺ -Distributivity) in E . Thus E is a basic poset.Conversely, say e ≺ f in E . Then we have g ∈ E with f ≺ g in S , as S is ≻≻ -round. In particular, f ≤ g so we have some e ′ ∈ E with e ′ ⊥ e and g ≤ e ′ ∨ f (noting joins in E are joins in S because S is a conditional ∨ -semilattice). Thus in S we have e ≤ f ≺ g ≤ e ′ ∨ f so e ≺ f , by ( ⊥ -Distributivity). (cid:3) Proposition 5.11. If S is a basic semigroup then a ≺ b, c ≺ d and b − b ` dd − ⇒ ac ≺ bd. Proof. If a, b, c, d ∈ S satisfy the given conditions, a − a ≺ b − b and cc − ≺ dd − ,by (Bi-Below). Applying ( ∧ -Preservation) to E yields a − acc − = a − a ∧ cc − ≺ b − b ∧ dd − = b − bdd − . By Proposition 5.4, acc − ≺ bdd − or, equivalently, cc − a − ≺ dd − b − . Thus acc − a − ≺ bdd − b − , by (Bi-Below), so ac ≺ bd , by Proposition 5.4. (cid:3) Corollary 5.12. If S is a basic semigroup then (Invariance) a ≺ b and b − b ≤ cc − ⇒ ac ≺ bc. Proof.
Assume a ≺ b and b − b ≤ cc − . By ( ≻ -Round), we have d ≻ c and hence dd − ≥ cc − ≥ b − b . As ≤ ⊆ ` , we have b − b ` dd − so ac ≺ bd = bc , byProposition 5.11. (cid:3) Groupoids
Topology.From now on we assume G is both a groupoid and a topological space . In other words, G is a topological space on which we have a partially definedmultiplication · which turns G into a small category in which every point is anisomorphism. Generally, we also want the topology to behave nicely with respectproducts and inverses, for which we recall the following standard definitions. Definition 6.1.
We call the groupoid G (1) topological if g g − and ( g, h ) gh are continuous.(2) ´etale if, moreover, g g − g is a local homeomorphism.First, we want to extend Proposition 2.18 to ‘bi-Hausdorff bisections’. Definition 6.2.
We denote the unit space of G by G . We call B ⊆ G (1) a bisection if B − B ⊆ G and BB − ⊆ G .(2) bi-Hausdorff if B − B and BB − are Hausdorff.Equivalently, B is a bisection iff g g − g and g gg − are injective on B .Indeed, if gh − ∈ G and g − g = h − h then g − g = g − gh − h = g − h so g = h .Note that we are not requiring bisections to be open. Proposition 6.3. If G is ´etale and locally Hausdorff then any compact bi-Hausdorffbisection C ⊆ G is contained in an open Hausdorff bisection. Proof.
For every g ∈ C , take some open Hausdorff bisection O g ∋ g . For every h ∈ C \ O g , we have open Hausdorff bisections N h ∋ g and M h ∋ h . As C is abi-Hausdorff bisection, g − g = h − h and gg − = hh − can be separated by opensets. As g g − g and g gg − are continuous, this means we can make N h and M h smaller if necessary to ensure that( N − h · N h ) ∩ ( M − h · M h ) = ∅ = ( N h · N − h ) ∩ ( M h · M − h )so N h ∪ M h is also an open Hausdorff bisection. As C is compact and O g is open, C \ O g is also compact so we have some finite subcover M ′ g = M h ∪ . . . ∪ M h k of C \ O g . Let N ′ g = O g ∩ N h ∩ . . . ∩ N h k . Again as C is compact, we have some finitesubcover N = N ′ g ∪ . . . ∪ N ′ g j of C . Let O = N ∩ ( O g ∪ M g ) ∩ . . . ∩ ( O g j ∩ M g j ) . Certainly O is an open subset containing C . To see that O is a Hausdorff bisection,take any g, h ∈ O . As g ∈ N , we have g ∈ N ′ g l for some l . We also have h ∈ O g l ∪ M ′ g l . If h ∈ O g l then, as g ∈ N ′ g l ⊆ O g l too and O g l is Hausdorff, wecan separate h and g with disjoint open sets. Moreover, as O g l is a bisection, h − h = g − g and hh − = gg − . If h ∈ M ′ g l then h ∈ M h j for some j so, as N ′ g l ∪ M h j ⊆ N h j ∪ M h j is a Hausdorff bisection, the same argument applies. As g and h were arbitrary, we are done. (cid:3) In topological groupoids, multiplication preserves compactness as long as G isHausdorff, which will be important later on in Proposition 6.11. Proposition 6.4. If G is topological and G is Hausdorff, the product C · D ⊆ G of any compact C, D ⊆ G is again compact.Proof. It suffices to show that any net ( g λ ) ⊆ C · D has a subnet ( g α ) convergingto some g ∈ C · D . But if ( g λ ) ⊆ C · D then we have nets ( c λ ) ⊆ C and ( d λ ) ⊆ D with g λ = c λ d λ . As C and D are compact, we have subnets ( c α ) and ( d α ) with c α → c ∈ C and d α → d ∈ D (find a convergent subnet ( c α ) first, then take afurther subnet so that ( d α ) converges too). Thus c − α c α → c − c and d α d − α → dd − .Moreover, c − α c α = d α d − α ∈ G , for all α , so we must have c − c = dd − , as G is Hausdorff and limits are unique in Hausdorff spaces. Thus cd is defined and g α = c α d α → cd ∈ C · D , as required. (cid:3) ´Etale Bases.Definition 6.5. We call a basis S of G an ´etale basis if, for all O, N ∈ S ,(1) O − ∈ S .(2) O · N ∈ S .(3) O − · O ⊆ G .We call S a ∪ -´etale basis if (3) is replaced by(3 ′ ) O ∪ N ∈ S ⇔ O ∪ N ⊆ B, for some compact bi-Hausdorff bisection B ⊆ G. Note that (1) and (3) imply that every element of an ´etale basis must be abisection. Likewise, as with ( ∪ ) for ∪ -bases, taking O = N in the ⇒ part of (3 ′ ),we see that every element of a ∪ -´etale basis is contained in a compact bi-Hausdorffbisection. Then the ⇐ part of (3 ′ ) ensures that S is a conditional ∨ -semilattice.By Proposition 6.3, we could also restate (3 ′ ) using ⋐ as(3 ′ ) O ∪ N ∈ S ⇔ O ∪ N ⋐ B, for some open Hausdorff bisection B ⊆ G. ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 33
Proposition 6.6. G has an ´etale basis if and only if G is an ´etale groupoid.Proof. If G is an ´etale groupoid then we claim that the collection S of all openbisections is an ´etale basis. To see this, note that if O is a bisection then so is O − .Also, as g g − is continuous it is also homeomorphism, begin an involution.Thus O − ∈ S whenever O ∈ S . Likewise, if O, N ∈ S then O · N is a bisectionwhich is also open, by [Sim17, Lemma 2.4.11], i.e. O · N ∈ S .Conversely, assume S is an ´etale basis of G . As O − ∈ S , for all O ∈ S , themap g g − is continuous. Whenever gh ∈ M ∈ S , we can take O ∈ S with g ∈ O . Then h ∈ O − · M ∈ S so gh ∈ O · ( O − · M ) ⊆ M . Thus multiplicationis also continuous and hence G is topological. As O · N ∈ S , for all O, N ∈ S ,multiplication is also an open map. Moreover, for any e ∈ G , we have some O ∈ S with e ∈ O . Then O − · O ∈ S and e ∈ O − · O ⊆ G , so G is open. Thus G is´etale, by [Res07, Theorem 5.18]. (cid:3) Any ´etale basis for G forms an inverse semigroup under pointwise operations.The resulting inverse semigroup relations have natural interpretations within G . Proposition 6.7. If S is an ´etale basis then, for any O, N ∈ S , O ≤ N ⇔ O ⊆ N. (6.1) O ∼ N ⇔ O ∪ N is a bisection . (6.2) Proof. (6.1) If O ⊆ N then, for each g ∈ O , we have g = gg − g ∈ O · O − · N so O ⊆ O · O − · N . Conversely, if f g − h is defined, for some f, g ∈ O and h ∈ N then, as g ∈ O ⊆ N and N is a bisection, we must have h = g .Likewise, f = g so f g − h = gg − g = g and hence O · O − · N ⊆ O . Thus O = O · O − · N , which means O ≤ N .(6.2) If O · N − and O − · N are idempotents in S then they must consist onlyof units of G , i.e. O · N − , O − · N ⊆ G . As O and N are bisections, wealso have O · O − , O − · O, N · N − , N − · N ⊆ G and hence( O ∪ N ) · ( O ∪ N ) − , ( O ∪ N ) − · ( O ∪ N ) ⊆ G , i.e. O ∪ N is a bisection. (cid:3) Proposition 6.8. If G has a ∪ -´etale basis S then G is a locally compact locallyHausdorff ´etale groupoid.Proof. By Proposition 6.6, G is an ´etale groupoid. By the O = N case of Definition 6.5(3 ′ ), we see that every O ∈ S is contained in a compact bi-Hausdorff bisection B .As G is topological, B and hence O must be Hausdorff. Indeed, for any distinct g, h ∈ B , g − g = h − h can be separated by disjoint open sets. This means g and h can be separated by disjoint open sets, as g g − g is continuous. As each g ∈ G is contained in some O ∈ S , G is locally compact locally Hausdorff, byProposition 2.19. (cid:3) Unfortunately, the converse does not hold in general.
Example 6.9.
We construct a locally compact locally Hausdorff ´etale groupoid G such that every ´etale basis of G contains a set with no compact extension.To construct such a G , consider the bug-eyed Cantor space C given in Example 2.10(the bug-eyed interval would also do), denoting the two ‘eyes’ by x and y . Let B = C ⊔ C ′ ⊔ C ′′ consist of three copies of C and consider the equivalence relation G ⊆ B × B defined as follows. First, G ∩ ( C × C ) ∩ ( C ′ × C ′ ) ∩ ( C ′′ × C ′′ ) = ∆ B ∩ ( C × C ) ∩ ( C ′ × C ′ ) ∩ ( C ′′ × C ′′ ) , i.e. the restriction of G to C , C ′ or C ′′ is just the equality relation. Also, for c ∈ C ,( c, c ′ ) ∈ G ⇔ c ∈ C \ { y } . (6.3) ( c, c ′′ ) ∈ G ⇔ c ∈ C \ { x } . (6.4) ( c ′ , c ′′ ) ∈ G ⇔ c ∈ C \ { x, y } . (6.5)This completely determines the equivalence relation G , which becomes a groupoidin a standard way. Moreover, G is topological as a subspace of B × B . One canalso check that G has the local homeomorphisms necessary to be ´etale.Now any ´etale basis S of G must include open sets O and N with( x, x ′ ) ∈ O ⊆ C × C ′ . ( y, y ′′ ) ∈ N ⊆ C × C ′′ . As S is ´etale, O − · N ∈ S . As x and y are not isolated in C , we have a sequence( c n ) ⊆ C \ { x, y } with c n → x or, equivalently, c n → y . Thus ( c n , c ′ n ) → ( x, x ′ ) so( c n , c ′ n ) is eventually in O . Likewise ( c n , c ′′ n ) is eventually in N , as ( c n , c ′′ n ) → ( y, y ′′ ).Thus ( c ′ n , c ′′ n ) = ( c ′ n , c n ) · ( c n , c ′′ n ) is eventually in O − · N . However, ( c ′ n , c ′′ n ) has nolimit in G , as (6.5) excludes all of the potential limits ( x ′ , x ′′ ), ( y ′ , x ′′ ), ( x ′ , y ′′ ) and( y ′ , y ′′ ). This means no subnet of ( c ′ n , c ′′ n ) has a limit in G either so O − · N is notcontained in any compact subset of G .In particular, the ⇒ part of Definition 6.5 (3 ′ ) fails, so G has no ∪ -´etale basis.Note too that the above G is ´etale and has a basis of compact open subsets,even though it does not have an ´etale basis of compact open subsets. Although asimpler example of such a G would be the bug-eyed Cantor space itself, seen as an´etale groupoid with trivial multiplication.The non-Hausdorff nature of Example 6.9 was crucial. Indeed, as long as theunit space G is Hausdorff, we do have a converse of Proposition 6.8. Remark 6.10.
If one really does want to work with general locally compact locallyHausdorff groupoids G where G may not be Hausdorff, this would likely still befeasible. On the one hand, one would replace Definition 6.5 (3 ′ ) with O ∪ N ∈ S ⇔ ( O ∪ N ) − · ( O ∪ N ) ⊆ B and ( O ∪ N ) · ( O ∪ N ) − ⊆ C, for some compact Hausdorff B, C ⊆ G . On the other hand, one would replace ( ≃ -Joins) with a ∨ b exists ⇔ a ∼ b and ∃ e, f, g, h ( a − a ≺ e ` f ≻ b − b and aa − ≺ g ` h ≻ bb − ) . Also ⋐ would now correspond to ≺≺ rather than ≺ . While somewhat messier,essentially the same duality could probably be obtained. However, G is Hausdorfffor most ´etale groupoids of interest to operator algebraists and we already havesufficient generality for these, by Proposition 6.11. Proposition 6.11. If G is ´etale and G is locally compact and Hausdorff then G has a ∪ -´etale basis S . ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 35
Proof.
Simply let S be the collection of open subsets of G contained in some (auto-matically bi-Hausdorff) compact bisection. As G is ´etale and G is locally compact, S is a basis of G . As g g − is a homeomorphism, S also satisfies Definition 6.5(1). As G is ´etale and G is Hausdorff, S also satisfies Definition 6.5 (2), byProposition 6.4. The definition of S also immediately yields Definition 6.5 (3 ′ ). (cid:3) As in Theorem 2.20, we obtain obtain general examples of ≃ -basic semigroupsfrom ∪ -´etale bases (the proof is essentially the same). Theorem 6.12.
Let S be a ∪ -´etale basis for some (necessarily) ´etale groupoid G .Then S is a ≃ -basic semigroup. Also, for any O, N ∈ S , O ≺ N ⇔ O ⋐ N. (6.6) O ` N ⇔ O ∪ N is Hausdorff . (6.7)It might seem reasonable to guess that ≺≺ and ≺ should coincide in any inversesemigroup which is a basic poset. We now give an elementary counterexample,showing that S really needs to be ≻≻ -round, not just ≻ -round, for this to hold. Example 6.13.
First let G be the space consisting of two copies of the unit interval,i.e. G = [0 , ⊔ [0 , ′ . We turn this into an ´etale groupoid by defining r · r = r ′ · r ′ = r and r · r ′ = r ′ · r = r ′ , for each r ∈ [0 ,
1] and corresponding r ′ ∈ [0 , ′ . Let S bethe collection of all open bisections O of G such that 0 ′ is not in the boundary of O , i.e. such that 0 ′ ∈ O ⇒ ′ ∈ O. Then S is an ∪ -´etale basis of G and hence a ≃ -basic semigroup. In particular, S isa basic poset, by Proposition 5.9 (and E is also a basic poset, by Proposition 5.10).Now consider the groupoid H = G \ { ′ } . The map φ ( O ) = O ∩ H takes S to aninverse semigroup T of open bisections of H . Moreover, φ is an order isomorphismand even a ∼ -isomorphism, although not a semigroup homomorphism, as φ ([0 , ′ · [0 , ′ ) = φ ([0 , , = (0 ,
1] = (0 , ′ · (0 , ′ = φ ([0 , ′ ) · φ ([0 , ′ ) . In particular, both T and its idempotents form basic posets too and T even stillsatisfies ( ≃ -Joins). However, T is not ≻≻ -round. Indeed, a = (0 , ′ is a maximalelement of T , so trivially a ≺ a , even though e e for e = aa − = a − a = (0 , e ⊆ [0 , ∋ ,
1] which contains 0.6.3.
Lenz Groupoids.
In the next section we will examine a natural groupoidstructure on ≺ -ultrafilters. But first, we consider general ≤ -filters. Throughout this subsection, let S be an arbitrary inverse semigroup. By [Len08, Theorem 3.1], the set of all ≤ -filters on S becomes an inverse semigroupunder the multiplication operation · given by T · U = ( T U ) ≤ . Restricting multiplication to the case when( T − T ) ≤ = ( U U − ) ≤ , we obtain the Lenz groupoid of S – see [Law98, § § U = U U − U , for any filter U . Indeed, U ⊆ U U − U holds for arbitrary U ⊆ S ,while the reverse inclusion follows from the following slightly more general result. Proposition 6.14. If T, U ⊆ S and U is a filter then (6.8) T − T ⊆ ( U U − ) ≤ ⇔ T − T U ⊆ U. Proof. If T − T ⊆ ( U U − ) ≤ then, for any a, b ∈ T and c ∈ U , we have a ′ , b ′ ∈ U with a ′ b ′− ≤ a − b . As U is a filter, we have c ′ ∈ U with c ′ ≤ a ′ , b ′ , c so c ′ = c ′ c ′− c ′ ≤ a ′ b ′− c ≤ a − bc and hence a − bc ∈ U . Thus T − T U ⊆ U . Conversely,if T − T U ⊆ U then, for any a, b ∈ T and c ∈ U , we have a − bc ∈ U so a − b ≥ a − bcc − ∈ U U − , i.e. a − b ∈ ( U U − ) ≤ . Thus T − T ⊆ ( U U − ) ≤ . (cid:3) Also note that, as long as T = ∅ and U is upwards closed,(6.9) ( T − T U ) ≤ = U ⇔ T − T U ⊆ U. Indeed, taking a ∈ T , we have U ⊆ ( a − aU ) ≤ ⊆ ( T − T U ) ≤ . Also ( T − T U ) ≤ ⊆ U implies T − T U ⊆ U while, conversely, T − T U ⊆ U implies ( T − T U ) ≤ ⊆ U ≤ = U .We also note that T − T U ⊆ U implies that the product ( T U ) ≤ can be calculatedfrom any single element of T . Proposition 6.15. If a ∈ T and T − T U ⊆ U then ( aU ) ≤ = ( T U ) ≤ .Proof. It suffices to show that ( aU ) ≤ ⊆ ( bU ) ≤ , for all a, b ∈ T . But as T − T U ⊆ U ,this follows from ( aU ) ≤ ⊆ ( bb − aU ) ≤ ⊆ ( bT − T U ) ≤ ⊆ ( bU ) ≤ . (cid:3) The following shows that a product in a filter can be used to express the filteritself as a product in the Lenz groupoid.
Proposition 6.16. If W is a filter with ab ∈ W then we have filters U = ( W b − ) ≤ and V = ( a − W ) with ( U − U ) ≤ = ( V V − ) ≤ and W = ( U V ) ≤ .Proof. As ab ∈ W , we have abb − a − ∈ W W − so ( abb − a − W ) ≤ = W , by (6.8)and (6.9) (taking T = { b − a − } ). Thus( bb − a − W ) ≤ = ( bb − a − aa − W ) ≤ = ( a − abb − a − W ) ≤ = ( a − W ) ≤ . Also ( W − W ) ≤ = ( b − a − W ) ≤ , by (6.8) and Proposition 6.15, from which weobtain ( bW − W ) ≤ = ( bb − a − W ) ≤ = ( a − W ) ≤ . This and W = W W − W yields( bW − W b − ) ≤ = ( bW − W W − W b − ) ≤ = ( a − W W − a ) ≤ , i.e. ( U − U ) ≤ = ( V V − ) ≤ . Thus ( U V ) ≤ = ( abb − V ) ≤ = ( abb − a − W ) ≤ = W ,again by (6.8) and Proposition 6.15. (cid:3) Proposition 6.17.
The following are equivalent, for any filters
T, U ⊆ S . (1) ( T − T ) ≤ = ( U U − ) ≤ . (2) T − T ∩ U U − = T − T U U − . (3) T U ∩ T ZU = ∅ , where Z = S \ ( T − T ∩ U U − ) ≤ .Proof. (1) ⇒ (2) If a ∈ T − T ∩ U U − then a = aa − a ∈ T − T T − T U U − = T − T U U − ,i.e. T − T ∩ U U − ⊆ T − T U U − whenever T (or U ) is a filter. Now say( T − T ) ≤ = ( U U − ) ≤ . In particular, U U − ⊆ ( T − T ) ≤ so T U U − ⊆ T ,by (6.8), and hence T − T U U − ⊆ T − T . Likewise, T − T U U − ⊆ U U − and hence T − T U U − ⊆ T − T ∩ U U − . ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 37 (2) ⇒ (1) If T − T U U − ⊆ ( T − T ) ≤ then, for any u ∈ U ,( T − T ) ≤ ⊆ ( T − T uu − ) ≤ ⊆ ( T − T U U − ) ≤ ⊆ ( T − T ) ≤ . Likewise if T − T U U − ⊆ ( U U − ) ≤ then, for any t ∈ T ,( U U − ) ≤ ⊆ ( t − tU U − ) ≤ ⊆ ( T − T U U − ) ≤ ⊆ ( U U − ) ≤ . Thus if T − T U U − ⊆ ( T − T ∩ U U − ) ≤ then( T − T ) ≤ = ( T − T U U − ) ≤ = ( U U − ) ≤ . (2) ⇒ (3) Say (3) fails, so we have ab = a ′ cb ′ for some a, a ′ ∈ T , b, b ′ ∈ U and c / ∈ ( T − T ∩ U U − ) ≤ . In fact we can assume a = a ′ and b = b ′ , as T and U are filters (taking a ′′ ∈ T with a ′′ ≤ a, a ′ and b ′′ ∈ U with b ′′ ≤ b, b ′ ,we obtain a ′′ b ′′ = a ′′ a ′′− abb ′′− b ′′ = a ′′ a ′′− a ′ cb ′ b ′′− b ′′ = a ′′ cb ′′ ). Then c ≥ a − acbb − = a − abb − so c ∈ ( T − T U U − ) ≤ . In particular, it followsthat T − T ∩ U U − = T − T U U − , i.e. (2) also fails.(3) ⇒ (2) Say (2) and hence (1) fails so ( T − T ) ≤ = ( U U − ) ≤ . W.l.o.g. we canassume ( T − T ) ≤ * ( U U − ) ≤ . As ( T − T ) ≤ = { t − t : t ∈ T } ≤ , we can take t ∈ T with t − t / ∈ ( U U − ) ≤ . For any u ∈ U , we have tu = tt − tu where t − t ∈ E \ ( U U − ) ≤ ⊆ S \ ( T − T ∩ U U − ) ≤ , i.e. (3) also fails. (cid:3) Ultrafilter Groupoids.
In Proposition 3.4 we showed how each point g ina space corresponds to an ultrafilter U g = { O ∈ S : g ∈ O } of elements in abasis S . In ´etale groupoids, multiplying points also corresponds to multiplying thecorresponding ultrafilters. Proposition 6.18. If S is a basis of an ´etale groupoid G then U gh = ( U g U h ) ⊆ ,i.e. { O ∈ S : gh ∈ O } = { O ∈ S : g ∈ M ∈ S, h ∈ N ∈ S and M N ⊆ O } , whenever gh is defined. Moreover, consider the following statements. (1) gh is defined. (2) T U g U h = ∅ . (3) ( U − g U g ) ⊆ = ( U h U − h ) ⊆ .Generally, (1) ⇒ (2) and (3) . If G is T , (1) ⇔ (3) . If G is T , (1) ⇔ (2) .Proof. If g ∈ M ∈ S , h ∈ N ∈ S and M N ⊆ O then certainly gh ∈ O , i.e.( U g U h ) ⊆ ⊆ U gh . Conversely, if gh ∈ O ∈ S then, as G is ´etale and S is a basis, wehave some bisection N ∈ U h , i.e. h ∈ N ∈ S so g = ghh − ∈ O · N − . As G is´etale, the product is an open map so O · N − is open. As S is a basis, we thus havesome M ∈ U g , i.e. g ∈ M ∈ S , such that M ⊆ O · N − . Then M · N ⊆ O · N − · N ⊆ O. As O was arbitrary, this shows that U gh ⊆ ( U g U h ) ⊆ , as required.(1) ⇒ (2) If gh is defined then gh ∈ T U g U h .(1) ⇒ (3) If gh is defined, g − g = hh − so ( U − g U g ) ⊆ = U g − g = U hh − = ( U h U − h ) ⊆ .(3) ⇒ (1) If gh is not defined then g − g = hh − , so if G is T then we have some O ∈ S distinguishing g − g and hh − , i.e. O is in precisely one of U g − g and U hh − and hence ( U − g U g ) ⊆ = U g − g = U hh − = ( U h U − h ) ⊆ . (2) ⇒ (1) Assume G is T . If gh is not defined then again g − g = hh − so we havesome O ∈ S with g − g ∈ O hh − . As G is ´etale, we have a bisection M ∈ U g with M − · M ⊆ O hh − . For any f ∈ M , f − f = hh − so,again as G is T , we have some O ′ ∈ S with hh − ∈ O ′ f − f . Then wehave a bisection N ∈ U h with N · N − ⊆ O ′ f − f . Thus { f } · N = ∅ , asthe product with f is not defined for any element of N . As M is a bisection,this means M · N does not contain any element of the form f k , for k ∈ N .As f was arbitrary, this shows that T U g U h ⊆ T { M · N : N ∈ U h } = ∅ . (cid:3) Our primary goal in this section is to show that we can also turn the ≺ -ultrafilterspace of a basic semigroup into an ´etale groupoid by multiplying ≺ -ultrafilters. Let S be a basic semigroup and let G be the set of ≺ -ultrafilters of S .Proposition 6.19. For any a ∈ S and U ∈ G , the following are equivalent. (1) aU ⊆ V ∈ G , for some V . (2) ( aU ) ≤ ∈ G . (3) a − a ∈ ( U U − ) ≤ . (4) a − a ∈ ( U U − ) ` and / ∈ bU , for some b ≺ a .Proof. (4) ⇒ (1) Say a − a ∈ ( U U − ) ` , 0 / ∈ bU and b ≺ a . So a − a ` uv − , for some u, v ∈ U . As U is ≤ -directed, we have some w ∈ U with w ≤ u, v so ww − ≤ uv − and hence a − a ` ww − , by ( ` -Auxiliarity).We claim V = S b ≺ c ≤ a cU ⊇ aU is ≺ -directed. As U is a ≺ -filter, wecan replace U with U w , i.e. it suffices to show { cu : b ≺ c ≤ a, w ≥ v ∈ U } is ≺ -directed. But { c : b ≺ c ≤ a } is ≺ -directed, by (Locally Hausdorff)and ( ` -Interpolation), and U w is ≺ -directed, as U is a ≺ -filter. ThusProposition 5.11, a − a ` ww − and ( ` -Auxiliarity) imply that their prod-uct is also ≺ -directed. Thus V can be extended to a ≺ -ultrafilter.(1) ⇒ (2) Assume aU ⊆ V ∈ G and take any u ∈ U . Then U ⊆ ( a − aU ) ≤ ⊆ ( a − V ) ≤ = ( a − V au ) ≤ . By (Invariance), a − V au is ≺ -directed so ( a − V au ) ≤ is a proper ≺ -filter.By the maximality of U , we have equality above, i.e. U = ( a − V ) ≤ so V ⊆ ( aa − V ) ≤ = ( a ( a − V ) ≤ ) ≤ = ( aU ) ≤ . By the maximality of V , equality holds again, i.e. ( aU ) ≤ = V ∈ G .(2) ⇒ (3) Assume V = ( aU ) ≤ ∈ G . Then as above, for any u ∈ U , U ⊆ ( a − aU ) ≤ ⊆ ( a − V ) ≤ = ( a − V au ) ≤ . The maximality of U again yields equality so U = ( a − aU ) ≤ . In particular, a − au ∈ U so a − a ≥ a − auu − a − a ∈ U U − , i.e. a − a ∈ ( U U − ) ≤ .(3) ⇒ (4) Assume a − a ∈ ( U U − ) ≤ ⊆ ( U U − ) ` , i.e. a − a ≥ uv − for some u, v ∈ U .Taking w ∈ U with w ≤ u, v , we have ww − ≤ a − a . Taking x ∈ U with x ≺ w , (Invariance) yields both xx − = xw − ≺ ww − ≤ a − a and b = axx − ≺ aa − a = a . For any u ∈ U , we have v ∈ U with v ≤ u, x andhence bu = axx − u ≥ avv − v = av . As v − a − av ≥ v − xx − v ≥ v − vv − v = v − v = 0 , we have av = 0 and hence bu = 0. (cid:3) ON-COMMUTATIVE LOCALLY COMPACT LOCALLY HAUSDORFF STONE DUALITY 39
In a similar vein, we have the following.
Corollary 6.20.
For any
U, V ∈ G , the following are equivalent. (1) U V ⊆ W ∈ G , for some W . (2) ( U − U ) ≤ = ( V V − ) ≤ (3) 0 / ∈ U V and u − u ` vv − , for some u ∈ U and v ∈ V .Proof. (3) ⇒ (1) As U and V are ≤ -filters, U V is ≤ -directed. If u − u ` vv − , for some u ∈ U and v ∈ V , then U V is even ≺ -directed. Indeed, for any u ′ ∈ U and v ′ ∈ V we have u ′′ , u ′′′ ∈ U and v ′′ , v ′′′ ∈ V with u ′′′ ≺ u ′′ ≤ u ′ , u and v ′′′ ≺ v ′′ ≤ v ′ , v so, by Proposition 5.11, u ′′′ v ′′′ ≺ u ′′ v ′′ ≤ u ′ v ′ . Thus wecan extend U V to some ≺ -ultrafilter W .(1) ⇒ (2) If U V ⊆ W ∈ G then, for any a, b ∈ U , we have c ∈ U with c ≤ a, b . Then cV ⊆ W so Proposition 6.19 yields a − b ≥ c − c ∈ ( V V − ) ≤ . As a and b were arbitrary, this shows that U − U ⊆ ( V V − ) ≤ . Likewise V V − ⊆ ( U − U ) ≤ and hence ( U − U ) ≤ = ( V V − ) ≤ .(2) ⇒ (3) If ( U − U ) ≤ = ( V V − ) ≤ then, for all u ∈ U , we have u − u ∈ ( V V − ) ≤ soProposition 6.19 yields u − u ∈ ( V V − ) ` and 0 / ∈ uV . As u was arbitrary,0 / ∈ U V . (cid:3) Proposition 6.21. G is a subgroupoid of the Lenz groupoid of S .Proof. We need to show that
U, V ∈ G and ( U − U ) ≤ = ( V V − ) ≤ ⇒ ( U V ) ≤ ∈ G. For any u ∈ U , ( U − U ) ≤ = ( V V − ) ≤ implies u − u ∈ ( V V − ) ≤ . As V ∈ G ,Proposition 6.19 then yields ( uV ) ≤ ∈ G . But ( U V ) ≤ = ( uV ) ≤ , by (6.8) andProposition 6.15. (cid:3) Theorem 6.22.
Let S be a basic semigroup and let G be the set of ≺ -ultrafiltersof S . Then G is a locally compact locally Hausdorff ´etale groupoid and O − a = O a − .O ab = O a · O b . Moreover, if S is a ≃ -basic semigroup then ( O a ) a ∈ S is an ∪ -´etale basis for G .Proof. By Theorem 3.9, G is locally compact locally Hausdorff.If a ∈ U and b ∈ V then certainly ab ∈ U V so O a · O b ⊆ O ab . Conversely, if ab ∈ W ∈ G then b ∈ ( a − W ) ≤ ∈ G , by Proposition 6.19, as aa − ≥ abb − a − ∈ W W − . Likewise, a ∈ ( W b − ) ≤ ∈ G . Setting U = ( W b − ) ≤ and V = ( a − W ) ≤ ,we have ( U − U ) ≤ = ( V V − ) ≤ and W = ( U V ) ≤ , by Proposition 6.16, and hence W ∈ O a · O b . As W was arbitrary, O ab ⊆ O a · O b . Thus ( O a ) a ∈ S is an ´etale basisand hence G is an ´etale groupoid, by Proposition 6.6.If S is a ≃ -basic semigroup then we can show that ( O a ) a ∈ S is a ∪ -´etale basis byessentially the same argument as at the end of the proof of Theorem 3.9. (cid:3) This completes the duality as far as the objects are concerned. As in §
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