aa r X i v : . [ m a t h . GN ] F e b GR ¨ATZER-HOFMANN-LAWSON-JUNG-S ¨UNDERHAUFDUALITY
TRISTAN BICE
Abstract.
We unify several extensions of the classic Stone duality due toGr¨atzer, Hoffman-Lawson and Jung-S¨underhauf. Specifically we show that ∪ -bases of locally compact sober spaces are dual to ≺ -distributive ∨ -predomains,where ≺ is a transitive relation representing compact containment. Introduction
Historical Background.
Over 80 years ago in [Sto36], Stone exhibited a re-markable duality between Boolean algebras and clopen bases of compact Hausdorffspaces, one which inspired a whole host of dualities between algebraic and more an-alytic structures. Even the specific duality that Stone examined would be extendedseveral times in the ensuing decades. The first step was taken by Stone himself in[Sto38], where he extended to the duality from Boolean algebras to general distribu-tive lattices, and from clopen bases of compact Hausdorff spaces to compact openbases of locally compact sober spaces that are coherent. It was only much later in[Gr¨a78, II.5] that coherence was eliminated by Gr¨atzer, the key idea being to workwith ∨ -semilattices satisfying an appropriate generalisation of distributivity.Around the same time, Hofmann and Lawson obtained a duality for generallocally compact sober spaces in [HL78], but at the cost of dealing with big lattices,namely continuous frames representing the entire open set lattice. It was only muchlater again in [JS96], building on ideas from [Smy92], that a partial unification ofthe Hofmann-Lawson and Stone dualities was achieved. However, there were stilla couple of issues, e.g. Jung-S¨underhauf’s work was again restricted to coherentspaces. Also, their axioms were too strong to apply to the entire open set lattice,and even finding an appropriate basis to satisfy their axioms turned out to be anon-trivial task.Nevertheless, the basic idea of [JS96] was perfectly sound, namely to work witha lattice, or better yet a ∨ -semilattice, together with an extra transitive relation ≺ representing compact containment. However, instead of imposing proximity-likeconditions on ≺ , we take leaf out of Gr¨atzer’s book and formulate an even moregeneral notion of distributivity for ≺ . The theory then proceeds more smoothlyalong classical lines (e.g. extending the Birkhoff-Stone prime filter theorem to ≺ ),allowing us to simultaneously generalise and hence unify all these Stone dualityextensions. Moreover, we make the duality functorial via certain relational mor-phisms, just like in [JS96] and [BS19a]. Mathematics Subject Classification.
Key words and phrases.
Stone duality, distributivity, semilattice, predomain, sober space.The author is supported by the GAˇCR project EXPRO 20-31529X and RVO: 67985840 at theInstitute of Mathematics of the Czech Academy of Sciences in Prague, Czech Republic.
Related Work.
In doing this, we bring Stone’s duality much closer to a contem-poraneous duality by Wallman in [Wal38]. Indeed, it is an interesting artefact ofhistory that Wallman’s duality was largely overlooked in favour of Stone’s duality,even though Wallman’s duality applies to (e.g. connected) spaces more commonlyseen in other fields. However, even the most general form of Stone duality pre-sented here differs in some important respects to Wallman’s duality. For example,the former deals with sober spaces while the latter deals with T spaces, which areincomparable notions even among compact spaces (see [PP12, I.3.1]). Also Wall-man’s duality applies to compact spaces, even those that are not locally compact(in the sense of points having compact neighbourhood bases). Recently in [BK20],we extended Wallman’s duality to ‘locally closed compact’ spaces which, in theHausdorff case, coincide with locally compact spaces. But even in the this case thedualities differ in that Wallman’s duality can be further generalised to semilatticesrepresenting mere subbases (again see [BK20]).In a different direction, the original Stone duality in [Sto36] can be generalisedby reducing the reliance on joins/unions. This is particularly important for obtain-ing non-commutative extensions to inverse semigroups and ´etale groupoids. Forexample, in [BS19a] we showed how to extend the classic Stone duality in [Sto36]to locally Hausdorff spaces (even non-zero dimensional ones) by considering con-ditional ∨ -semilattices. Even without any joins whatsoever, a duality can be stillobtained with bases or even pseudobases of locally compact Hausdorff spaces. Thiswe showed in [BS19b], providing a common generalisation of Exel’s tight groupoidconstruction from [Exe08] and the earlier Stone duality extensions of Shirota [Shi52]and de Vries [Vri62]. Note, however, that without joins we are forced to rely moreheavily on stronger separation properties, like being (locally) Hausdorff. Outline.
First in §
1, we examine the spectrum of proper round prime filters inany ∨ -semilattice S with minimum 0 together with an additional relation ≺ . Thekey results are that the spectrum is always sober – see Proposition 1.4 – andthat the spectrum recovers any core compact sober space from any ∪ -basis – seeTheorem 1.6.Next, in §
2, we investigate various properties of ≺ and their relation to thespectrum. Most of these properties have been considered before in domain theory– see [GHK + ≺ , a variant of a previous version in [BS19a] – see (Distributivity).This allows for a direct generalisation of the Birkhoff-Stone prime filter theorem inTheorem 2.4. Together with the conditions defining ∨ -predomains, it then followsthat the spectrum is core compact and that ≤ and ≺ are faithfully represented onthe spectrum as ⊆ and ⋐ respectively – see Corollary 2.8, (2.1) and (2.2).Finally, in §
3, we make the duality functorial with respect to certain relationalmorphisms defined by a few simple first order properties – see Definition 3.1. Witha couple of extra conditions on the relational morphisms, this even yields an equiv-alence between categories of ≺ -distributive ∨ -predomains and ∪ -bases of locallycompact sober spaces. R¨ATZER-HOFMANN-LAWSON-JUNG-S ¨UNDERHAUF DUALITY 3 The Spectrum
We make the following standing assumption throughout the paper.
We have a relation ≺ on a ∨ -semilattice S with minimum . First we recall the following standard definitions from order theory.
Definition 1.1.
We call F ⊆ S a filter and I ⊆ S an ideal if p, q ∈ F ⇔ ∃ r ∈ F ( r ≤ p, q ) . (Filter) p, q ∈ I ⇔ ∃ r ∈ I ( r ≥ p, q ) . (Ideal)A filter P ⊆ S is prime if S \ P is an ideal. As S is a ∨ -semilattice, this means p ∨ q ∈ P ⇒ p ∈ P or q ∈ P. (Prime)A filter R ⊆ S is round if p ∈ R ⇒ ∃ r ∈ R ( r ≺ p ) . (Round) Definition 1.2.
The spectrum of ( S, ≤ , ≺ ) is the space with points b S = { P ⊆ S : P is a proper round prime filter } with the topology generated by ( b S p ) p ∈ S where b S p = { P ∈ b S : p ∈ P } . Note that, as b S consists of filters, ( b S p ) p ∈ S is a basis for the topology on b S . Definition 1.3.
A closed set C ⊆ X is irreducible if its proper closed subsets forman ideal. A space X is sober if every irreducible C has a unique dense point x , i.e. C = cl { x } . Proposition 1.4.
The spectrum is always sober.Proof.
Take any irreducible C ⊆ b S and let P = { p ∈ S : b S p ∩ C = ∅} . If q ≥ p ∈ P then b S q ∩ C ⊇ b S p ∩ C = ∅ and hence q ∈ P too. Also, for any p, q ∈ P , b S p ∩ C = ∅ 6 = b S q ∩ C implies b S p ∩ b S q ∩ C = ∅ , as C is irreducible. Taking any Q ∈ b S p ∩ b S q ∩ C , we see that p, q ∈ Q and hence r ∈ Q , for some r ≤ p, q , as Q is afilter. Thus Q ∈ b S r ∩ C = ∅ and hence r ∈ P , showing that P is a filter. Similarly,for any p ∈ P , we have Q ∈ b S p ∩ C so p ∈ Q and hence we have q ∈ Q with q ≺ p ,as Q is round. Thus Q ∈ b O q ∩ C = ∅ so q ∈ P , showing that P is also round. Forany p, q ∈ S \ P , we have b S p ∨ q = b S p ∪ b S q ⊆ X \ C so S \ P is an ideal. Also b S = ∅ so 0 / ∈ P and hence P is a proper round prime filter, i.e. P ∈ b S .As P / ∈ b S q implies q / ∈ P and hence b S q ∩ C = ∅ , it follows that C ⊆ cl { P } . Onthe other hand, as b S p ∩ C = ∅ , for all p ∈ P , we have P ∈ cl( C ) = C and hence P isa dense point in C . Moreover, P is unique, as b S is immediately seen to be T . (cid:3) TRISTAN BICE
Conversely, any ‘core compact’ sober space arises in this way. Specifically, forany topology O ( X ) on X , let ⋐ denote the way-below relation on O ( X ), i.e. p ⋐ q means that every open cover of q has a finite subcover of p : p ⋐ q ⇔ ∀C ⊆ O ( X ) ( q ⊆ [ C ⇒ ∃ finite
F ⊆ C ( p ⊆ [ F )) . As in [GL13, § X core compact if O ( X ) is a continuous frame, i.e. ifevery open neighbourhood filter O x = { O ∈ O ( X ) : x ∈ O } is round w.r.t. ⋐ . Definition 1.5. A ∪ -basis S ⊆ O ( X ) is a basis closed under finite unions.We include the empty union here, i.e. the empty set ∅ = S ∅ . So, equivalently,a ∪ -basis is a basis containing ∅ which is closed under pairwise unions. Theorem 1.6.
For any ∪ -basis S of a core compact sober space X , x S x = { s ∈ S : x ∈ s } is a homemorphism from X onto b S , where ( S, ≤ , ≺ ) = ( S, ⊆ , ⋐ ) .Proof. Each S x is certainly a proper prime filter (proper because ∅ ∈ S \ S x ). As X is core compact, each S x is also round and hence S x ∈ b S .Conversely, say P ∈ b S . Let O = S ( S \ P ) and note that p * O , for all p ∈ P – ifwe had p ⊆ O then, as P is round, we would have q ∈ P with q ⋐ p ⊆ O and hence q ⊆ S F , for some finite F ⊆ S \ P , contradicting the fact P is prime. For anyopen N % O = S ( S \ P ), we have p ∈ P with p ⊆ N , as S is a basis. For any other N ′ % O , we again have p ′ ∈ P with p ′ ⊆ N ′ . As P is a filter, we have p ′′ ∈ P with p ′′ ⊆ p ∩ p ′ ⊆ N ∩ N ′ . Thus N ∩ N ′ ⊇ p ′′ * O , as p ′′ ∈ P , so O = N ∩ N ′ , showingthat X \ O is irreducible. As X is sober, we have x ∈ X with X \ O = cl { x } . As x / ∈ O = S ( S \ P ), S x ⊆ P . On the other hand, if s ∈ S \ S x then s ∩ cl { x } = ∅ and hence s ⊆ O , which can only mean s / ∈ P . Thus S x = P .So x S x is a bijection from X to b S . Consequently, it is a homeomorphism, aswe immediately see that it maps the basis S onto the basis ( b S p ) p ∈ S . (cid:3) Our next goal is to isolate the abstract properties of ( S, ≤ , ≺ ) that make b S corecompact and ensure that ≺ and ≤ are faithfully represented as ⊆ and ⋐ on ( b S p ) p ∈ S . Remark 1.7.
A more standard approach would be to take ( S, ≤ ) as the officialstructure and derive ≺ in some way from ≤ , e.g. if ( S, ≤ , ≺ ) = ( O ( X ) , ⊆ , ⋐ ) then ≺ is, by definition, the way-below relation derived from ≤ on S . However, we aremore interested in ∪ -bases S ⊆ O ( X ). In this case, as long as X is Hausdorff andeach s ∈ S is relatively compact, ≺ can instead be derived as the rather belowrelation from ≤ (see [BS19a]). But in more general sober spaces, even the compactand core compact ones, there may actually be no way of defining ≺ from ≤ .To see this, consider S = ω + 2 = { , , . . . , ω, ω + 1 } with its usual ordering ≤ .On the one hand, we could take ≺ = ≤ too and then ( S, ≤ , ≺ ) would represent the ∪ -basis ( ω + 3) \ { ω } of X = ω + 2 in its lower topology. On the other hand, if wemodify ≺ just a little by declaring that ω ω then ( S, ≤ , ≺ ) would represent theentire open set lattice of X = ω + 1 in its lower topology.A better idea for our work would be to instead take ( S, ≺ ) as the official structureand derive ≤ as the canonical lower preorder of ≺ – see (Lower Preorder) below.We will do this eventually, but initially it suffices to consider two separate relations ≺ and ≤ that are simply related in various ways. R¨ATZER-HOFMANN-LAWSON-JUNG-S ¨UNDERHAUF DUALITY 5 ≺ -Distributive ∨ -Predomains Definition 2.1.
We call ≺ distributive if, for all p, s, t ∈ S ,(Distributivity) p ≺ s ∨ t ⇒ ∀ p ′ ≺ p ∃ s ′ ≺ s ∃ t ′ ≺ t ( p ′ ≤ s ′ ∨ t ′ ≤ p ) . Note that when ≺ = ≤ , it suffices to take p ′ = p above, which then reduces tothe usual notion of distributivity for ∨ -semilattices.The motivating example we have in mind is ⋐ . Proposition 2.2. If S is a ∪ -basis of a core compact space X then ⋐ is distributive.Proof. Say p ′ ⋐ p ⊆ s ∪ t . For every x ∈ p , we have x ∈ s or x ∈ t . As S is abasis and X is core compact, this means we have q ∈ S with either x ∈ q ⋐ p ∩ s or x ∈ q ⋐ p ∩ t . As p ′ ⋐ p , finitely many such q cover p ′ . As S is ∪ -closed, theunion of those q contained in s yields s ′ ⋐ s and the union of those q contained in t yields t ′ ⋐ t such that p ′ ⊆ s ′ ∪ t ′ ⊆ p . (cid:3) Another important property of ⋐ is auxiliarity. Definition 2.3.
We call ≺ auxiliary if, for all p, p ′ , q, q ′ ∈ S ,(Auxiliarity) p ≤ p ′ ≺ q ′ ≤ q ⇒ p ≺ q ⇒ p ≤ q. Note that when ≺ is auxiliary, round filters are precisely the ≺ -filters , i.e.( ≺ -Filter) p, q ∈ F ⇔ ∃ r ∈ F ( r ≺ p, q ) , which means that F is ≺ -directed and upwards closed w.r.t. ≺ , i.e.(Up-Set) F ⊆ F ≺ = { p ∈ S : ∃ f ∈ F ( f ≺ p ) } . When ≺ is also distributive, we can extend the Birkhoff-Stone prime filter theorem. Theorem 2.4. If ≺ is distributive and auxiliary, I ⊆ S is an ideal and F ⊆ S is around filter with I ∩ F = ∅ then F extends to a round prime filter P with I ∩ P = ∅ .Proof. By Kuratowski-Zorn, we can extend I to an ideal J that is maximal amongideals disjoint from F . Then P = S \ J is an up-set and we claim that P isalso a round filter. To see this, take p, q ∈ P . By maximality, we have j, k ∈ J with p ∨ j ∈ F and q ∨ k ∈ F . As F is a round filter, we have f , · · · , f ∈ F with f ≺ · · · ≺ f ≤ p ∨ j, q ∨ k . As ≺ is auxiliary, f ≺ p ∨ j, q ∨ k . As ≺ is distributive, wehave p ′ ≺ p and j ′ ≺ j with f ≤ p ′ ∨ j ′ ≤ f . Applying auxiliarity and distributivityagain, we get p ′′ ≺ p ′ and j ′′ ≺ j ′ with f ≤ p ′′ ∨ j ′′ ≤ f . As p ′′ ≺ p ′ ≤ f ≺ q ∨ k ,distributivity yet again yields q ′ ≺ q and k ′ ≺ k with p ′′ ≤ q ′ ∨ k ′ ≤ p ′ . Note q ′ ≺ q and q ′ ≤ p ′ ≺ p . Also j ∨ k ∈ J and j ∨ k ∨ q ′ ≥ j ′ ∨ k ′ ∨ q ′ ≥ j ′′ ∨ p ′′ ≥ f ∈ F sowe must have q ′ ∈ P . This shows that P is a ≺ -filter and hence a round filter. As J = S \ P is an ideal, P is also prime. (cid:3) Definition 2.5.
We say ≺ is interpolative if, for all p, q ∈ S ,(Interpolation) p ≺ q ⇒ ∃ s ( p ≺ s ≺ q ) . Again, the key example to keep in mind is ⋐ . Proposition 2.6. If S is a ∪ -basis of core compact X then ⋐ has interpolation.Proof. Say p, q ∈ S satisfy p ⋐ q . As X is core compact, for any x ∈ q , we have r, s ∈ S with x ∈ r ⋐ s ⋐ q . As p ⋐ q , finitely many such r cover p . Taking theunion of these r ’s and s ’s yields r ′ , s ′ ∈ S with p ⊆ r ′ ⋐ s ′ ⋐ q , as required. (cid:3) TRISTAN BICE
Proposition 2.7. If ≺ is distributive, auxiliary and interpolative then p ≺ q ⇒ ∃ compact C ⊆ b S ( b S p ⊆ C ⊆ b S q ) . Proof.
By interpolation, we have ≺ -filter F ⊆ p ≺ containing q (take the upwardsclosure of ( p n ) where p ≺ p k +1 ≺ p k ≺ q , for all k ). We claim that C = T f ∈ F b S f iscompact, which proves the result, as b S p ⊆ C ⊆ b S q . For this it suffices to show ∀ j ∈ I ( C * b S j ) ⇒ C * [ j ∈ I b S j , for any non-empty ideal I ⊆ S . But C * b S j implies j / ∈ F , so if this holds for all j ∈ I then I ∩ F = ∅ . Then Theorem 2.4 yields P ∈ b S extending F disjoint from I so P ∈ C \ S j ∈ I b S j witnesses C * S j ∈ I b S j , as required. (cid:3) Corollary 2.8. If ≺ is distributive, auxiliary and interpolative, b S is core compact.Proof. If P ∈ b S p then q ≺ p , for some q ∈ P . By Proposition 2.7, P ∈ b S q ⋐ b S p . (cid:3) Actually, what this really shows is that every point in b S has a compact neighbour-hood base. By Theorem 1.6, together with Proposition 2.2 and Proposition 2.6, thisprovides an alternative proof that core compact sober spaces are necessarily locallycompact (see [GL13, Theorem 8.3.10]).So Corollary 2.8 tells us that the spectrum produces the spaces we want. Theremaining question is what additional conditions ensure that ⊆ and ⋐ on the basis( b S p ) p ∈ S defined by S correspond precisely to ≺ and ≤ on S itself. Definition 2.9.
We call ≺ approximating if ≤ is the lower preorder of ≺ , i.e.(Lower Preorder) p ≤ q ⇔ p ≻ ⊆ q ≻ . Note that ⋐ is approximating on any basis of a core compact space. Remark 2.10.
We could have also used the original version of distributivity p ≤ s ∨ t ⇔ ∀ p ′ ≺ p ∃ s ′ ≺ s ∃ t ′ ≺ t ( p ′ ≺ s ′ ∨ t ′ ≺ p )from [BS19a], which already implies that ≺ is approximating and has interpolation.We avoided this in order to show that Theorem 2.4 only requires the weaker versionin (Distributivity) above. Proposition 2.11. If ≺ is distributive, auxiliary, interpolative and approximating, (2.1) p ≤ q ⇔ b S p ⊆ b S q . Proof.
The ⇒ part is immediate. Conversely, say p (cid:2) q . As ≺ is approximating,we have r ≺ p with r q . As ≺ has interpolation, we have a ≺ -filter F ⊆ r ≺ containing p . As r q , F is disjoint from the ideal q ≥ . By Theorem 2.4, F extendsto a round prime filter P avoiding q so P ∈ b S p \ b S q witnesses b S p * b S q . (cid:3) Definition 2.12.
We say ≺ is ∨ -preserving if, for all p, p ′ , q, q ′ ∈ S , p ′ ≺ p and q ′ ≺ q ⇒ p ′ ∨ q ′ ≺ p ∨ q.S is a ∨ -predomain if ≺ is auxiliary, approximating, interpolative and ∨ -preserving.The ‘predomain’ terminology is due to Keimel – see [Kei17]. We immediately seethat ⋐ is ∪ -preserving on any ∪ -basis. Thus any ∪ -basis S ⊆ O ( X ) of core compact X is a ⋐ -distributive ∪ -predomain (see Proposition 2.2 and Proposition 2.6). R¨ATZER-HOFMANN-LAWSON-JUNG-S ¨UNDERHAUF DUALITY 7
Proposition 2.13. If S is a ≺ -distributive ∨ -predomain then, for any p, q ∈ S , (2.2) p ≺ q ⇔ b S p ⋐ b S q . Proof.
The ⇒ part is immediate from Proposition 2.7. Conversely, say p q . As ≺ preserves ∨ , q ≻ is an ideal and hence ( b S r ) r ≺ q generates an ideal of open setscovering b S q . However, none of these open sets covers b S p as b S p ⊆ b S r would imply p ≤ r ≺ q , by (2.1) and (2.2). Thus b S p ⋐ b S q . (cid:3) These results can be summarised as giving us a duality of the following classes. ∪ B = { S : S is a ∪ -basis of a core compact sober space } . ≺ D ∨ P = { S : S is a ≺ -distributive ∨ -predomain } . Theorem 2.14. ∪ B is dual to ≺ D ∨ P . More precisely, ∪ B ⊆ ≺ D ∨ P , by Proposition 2.2 and Proposition 2.6, i.e. any ∪ -basis of a core compact sober space is a ⋐ -distributive ∪ -predomain. Moreover, thespectrum of such a ∪ -basis recovers the original space, by Theorem 1.6. Conversely,every ≺ -distributive ∨ -predomain S ∈ ≺ D ∨ P has a core compact sober spectrum b S , by Proposition 1.4 and Corollary 2.8. Moreover, ≤ and ≺ become ⊆ and ⋐ on( b S p ) p ∈ S ∈ ∪ B , by (2.1) and (2.2), thus yielding a ∪ -basis isomorphic to S . Remark 2.15.
The above construction gives us a way of embedding any ≺ -distributive ∨ -predomain S into a continuous frame, all we have to do is note that S is isomorphic to ( b S p ) p ∈ S which is a subset of the continuous frame of all open sets O ( b S ). Alternatively, the continuous frame could be constructed directly from theround ideals – see [GL13, Proposition 5.1.33] or [GHK +
03, Exercise III.4.17] – itwould just have to be verified that ≺ -distributivity in S yields infinite distributivityof the round ideals. In [HL78], Hofmann-Lawson already showed that continuousframes are dual to core compact sober spaces, so this would also provide a some-what indirect way of obtaining our duality. Conversely, Hofmann-Lawson dualitycould be viewed as a special case of our duality when S is a complete lattice onwhich ≺ is defined as the way-below relation. Remark 2.16.
Gr¨atzer’s generalisation of Stone duality in [Gr¨a78, II.5] is also aspecial case of our duality when ≺ = ≤ . Incidentally, as we are already workingwith bases, one might think it would be harmless to deal with lattices rather thansemilattices. However, taking ≺ = ≤ in this case would force the resulting spacesto be coherent and we would only recover Stone’s duality for distributive lattices,rather than Gr¨atzer’s duality for distributive semilattices. Remark 2.17.
We also get Jung-S¨underhauf’s duality in [JS96] as a special casewhen ≺ satisfies their strong proximity axioms. This points to a potential appli-cation of our duality – just as proximity relations on subsets of a given space areoften used to obtain Tychonoff compactifications, distributive predomain relationscould be used to obtain more general sober (even local) compactifications.Another important way of representing bounded distributive lattices is on Priest-ley spaces (see [Pri70]), which are certain kinds of ordered Stone spaces. It wouldseem reasonable to guess that ≺ -distributive ∨ -predomains could also be repre-sented in a similar way on some class of (potentially connected) pospaces. Question.
Does Priestley duality extend to ≺ -distributive ∨ -predomains? TRISTAN BICE Morphisms
Note ∪ B becomes a category when we take partial continuous functions as mor-phisms, i.e. given S, S ′ ∈ ∪ B , a morphism from S to S ′ is a function φ on (neces-sarily open) dom( φ ) such that, for all p ′ ∈ S ′ , φ − [ p ′ ] = [ { p ∈ S : p ⊆ φ − [ p ′ ] } . Then φ yields a relation ⊏ φ ⊆ S × S ′ given by p ⊏ φ p ′ ⇔ p ⋐ φ − [ p ′ ] . Our goal is to axiomatise such relations in order to make our duality functorial. Atfirst, we will consider axioms that apply even when we replace ⋐ with ⊆ , i.e. p ⊏ p ′ ⇔ p ⊆ φ − [ p ′ ] . Definition 3.1.
Given
S, S ′ ∈ ≺ D ∨ P , we call ⊏ ⊆ S × S ′ a morphism if p ⊏ ′ ⇒ p = 0 . (Faithful) p ≤ q ⊏ q ′ ≤ p ′ ⇒ p ⊏ p ′ . (Auxiliarity) p ≺ q ⊏ r ′ , s ′ ⇒ ∃ p ′ ∈ S ′ ( p ⊏ q ′ ≺ r ′ , s ′ ) . (Pushforward) p ≺ q ⊏ r ′ ∨ s ′ ⇒ ∃ q ⊏ q ′ ∃ r ⊏ r ′ ( p ≺ r ′ ∨ s ′ ) . ( ∨ -Pullback)We consider the usual composition of relations, i.e. if ⊏ ⊆ S × S ′ and ⊏ ′ ⊆ S ′ × S ′′ , p ⊏ ◦ ⊏ ′ p ′′ ⇔ ∃ p ′ ∈ S ′ ( p ⊏ p ′ ⊏ p ′′ ) . Proposition 3.2. ≺ D ∨ P forms a category.Proof. It follows from (Auxiliarity) that each S ∈ ≺ D ∨ P has an identity morphism,namely ≤ . We just have to show that ⊏ ◦ ⊏ ′ is a morphism whenever ⊏ and ⊏ ′ are. We verify the required properties as follows.(Faithful) If p ⊏ p ′ ⊏ ′ ′′ then p ′ = 0 ′ and hence p = 0, by (Faithful) for ⊏ and ⊏ ′ .(Auxiliarity) If p ≤ q ⊏ p ′ ⊏ ′ q ′′ ≤ p ′′ then p ⊏ p ′ ⊏ ′ p ′′ , by (Auxiliarity) for ⊏ and ⊏ ′ .(Pushforward) If p ≺ q ⊏ q ′ ⊏ ′ r ′′ , s ′′ then (Pushforward) for ⊏ yields p ′ ∈ S ′ with p ⊏ p ′ ≺ q ′ ⊏ ′ r ′′ , s ′′ and then (Pushforward) for ⊏ ′ yields q ′′ ∈ S ′′ with p ⊏ p ′ ⊏ q ′′ ≺ r ′′ , s ′′ .( ∨ -Pullback) If p ≺ q ⊏ q ′ ⊏ ′ r ′′ ∨ s ′′ then ( ∨ -Pullback) for ⊏ ′ yields r ′ ⊏ ′ r ′′ and s ′ ⊏ ′ s ′′ with p ≺ q ⊏ q ′ ≺ r ′ ∨ s ′ and hence p ≺ q ⊏ r ′ ∨ s ′ , by (Auxiliarity) for ⊏ .Then ( ∨ -Pullback) for ⊏ yields r ⊏ r ′ and s ⊏ s ′ with p ≺ r ∨ s . (cid:3) Given ⊏ ⊆ S × S ′ define φ ⊏ from b S to b S ′ by φ ⊏ ( P ) = P ⊏ on dom( φ ) = { P ∈ b S : P ⊏ = ∅} . Proposition 3.3.
We have a functor from ≺ D ∨ P to ∪ B given by S ( b S p ) p ∈ S and ⊏ φ ⊏ . Proof.
For any S ∈ ≺ D ∨ P , we immediately see that φ ≤ is the identity map on b S .The main task is to show ran( φ ⊏ ) ⊆ b S ′ , for any morphism ⊏ between S, S ′ ∈ ≺ D ∨ P , i.e. that P ⊏ ∈ b S ′ whenever P ∈ b S and P ⊏ = ∅ . By (Faithful), P ⊏ isproper. Whenever P ∋ p ⊏ p ′ ≤ q ′ , (Auxiliarity) yields p ⊏ q ′ , showing that P ⊏ is an up-set. Whenever P ∋ p ⊏ r ′ , s ′ , the roundness of P yields t ∈ P with t ≺ p and then (Pushforward) yields p ′ ∈ S ′ with t ⊏ p ′ ≺ r ′ , s ′ , showing that P ⊏ is a R¨ATZER-HOFMANN-LAWSON-JUNG-S ¨UNDERHAUF DUALITY 9 ≺ -filter. Whenever P ∋ p ⊏ r ′ ∨ s ′ , the roundness of P again yields t ∈ P with t ≺ p and then ( ∨ -Pullback) yields r ⊏ r ′ and s ⊏ s ′ with t ≺ r ∨ s . As P is prime,either r ∈ P and hence r ′ ∈ P ⊏ or s ∈ P and hence s ′ ∈ P ⊏ , showing that P ⊏ isprime and hence P ⊏ ∈ b S ′ . Continuity then follows as, for all p ′ ∈ S ′ , φ − ⊏ [ b S ′ p ′ ] = [ p ⊏ p ′ b S p . Also composition is preserved as, for all P ∈ dom( φ ⊏ ◦ ⊏ ′ ) = φ − ⊏ [dom( φ ⊏ ′ )], φ ⊏ ◦ ⊏ ′ ( P ) = P ⊏ ◦ ⊏ ′ = ( P ⊏ ) ⊏ ′ = φ ⊏ ′ ( φ ⊏ ( P )) = φ ⊏ ′ ◦ φ ⊏ ( P ) . (cid:3) On the other hand, the map φ ⊏ φ is not (part of) a functor to ≺ D ∨ P , as itdoes not preserve identity morphisms. However, it does preserve composition. Proposition 3.4.
For any morphisms φ from S to S ′ and φ ′ from S ′ to S ′′ in ∪ B , ⊏ φ ′ ◦ φ = ⊏ φ ◦ ⊏ φ ′ . Proof.
Say p ⊏ φ ′ ◦ φ p ′′ , which means p ⋐ φ − [ φ ′− [ p ′′ ]] and hence we have compact C with p ⊆ C ⊆ φ − [ φ ′− [ p ′′ ]]. Thus φ [ C ] is also compact and φ [ C ] ⊆ φ ′− [ p ′′ ]. As S ′ is a ∪ -basis, this means we have p ′ ∈ S ′ with φ [ C ] ⊆ p ′ ⋐ φ ′− [ p ′′ ] and hence p ⊏ φ p ′ ⊏ φ ′ p ′′ . The converse is immediate from the definitions. (cid:3) Proposition 3.5.
For any morphism φ from S to S ′ in ∪ B , we always have S ′ φ ( x ) = φ ⊏ φ ( S x ) . Proof.
Just note that p ′ ∈ S ′ φ ( x ) iff, for some p ∈ S , x ∈ p ⋐ φ − [ p ′ ] which means S x ∋ p ⊏ φ p ′ , i.e. p ′ ∈ φ ⊏ φ ( S x ). (cid:3) For any ⊏ ⊆ S × S ′ , we define ⊏ ∨ ⊆ S × S ′ by p ⊏ ∨ p ′ ⇔ ∃ finite F ⊏ p ′ ( p ≺ _ F ) . Proposition 3.6.
For any morphism ⊏ between S, S ′ ∈ ≺ D ∨ P , p ⊏ ∨ p ′ ⇔ b S p ⊏ φ ⊏ b S ′ p ′ . Proof.
Say p ⊏ ∨ p ′ , so we have finite F ⊏ p ′ with p ≺ W F and hence b S p ⋐ b S W F , by(2.2). But F ⊏ p ′ implies b S W F = S f ∈ F b S f ⊆ φ − ⊏ [ b S ′ p ′ ] and hence b S p ⋐ φ − ⊏ [ b S ′ p ′ ],i.e. b S p ⊏ φ ⊏ b S ′ p ′ .Conversely, say b S p ⋐ φ − ⊏ [ b S ′ p ′ ]. Take q ∈ S with b S p ⋐ b S q ⋐ φ − ⊏ [ b S ′ p ′ ]. Thedefinition of φ ⊏ means we can cover φ − ⊏ [ b S ′ p ′ ] by sets b S f such that f ⊏ p ′ . Thenthe definition of ⋐ means we have finite F ⊏ p ′ with b S p ⋐ b S q ⊆ S f ∈ F b S f = b S W F and hence p ≺ W F , by (2.2), i.e. p ⊏ ∨ p ′ . (cid:3) These results show that we can obtain a category equivalent to ∪ B if we restrictto ‘ ∨ -morphisms’ in ≺ D ∨ P , i.e. those morphisms with ⊏ = ⊏ ∨ . Equivalently, theseare the morphisms satisfying the following extra conditions. p ⊏ p ′ ⇒ ∃ q ∈ S ( p ≺ q ⊏ p ′ ) . (Left Interpolation) q, r ⊏ p ′ ⇒ q ∨ r ⊏ p ′ . ( ∨ -Preserving) Note each S still has an identity in this new category, it is just ≺ instead of ≤ .Incidentally, if we wanted to restrict to total functions in ∪ B , then we would addone further condition on the morphisms in ≺ D ∨ P , namely(Total) p ≺ q ⇒ ∃ q ′ ∈ S ′ ( p ⊏ q ′ ) . References [BK20] Tristan Bice and Wies law Kubi´s. Wallman duality for semilattice subbases, 2020. arXiv:2002.05943 .[BS19a] Tristan Bice and Charles Starling. General non-commutative locallycompact locally Hausdorff Stone duality.
Adv. Math. , 314:40–91, 2019. doi:10.1016/j.aim.2018.10.031 .[BS19b] Tristan Bice and Charles Starling. Hausdorff tight groupoids generalised.
SemigroupForum , 2019. doi:10.1007/s00233-019-10027-y .[Exe08] Ruy Exel. Inverse semigroups and combinatorial C ∗ -algebras. Bull. Braz. Math. Soc.(N.S.) , 39(2):191–313, 2008. doi:10.1007/s00574-008-0080-7 .[GHK +
03] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S.Scott.
Continuous lattices and domains , volume 93 of
Encyclopedia of Math-ematics and its Applications . Cambridge University Press, Cambridge, 2003. doi:10.1017/CBO9780511542725 .[GL13] Jean Goubault-Larrecq.
Non-Hausdorff topology and domain theory , volume 22 of
New Mathematical Monographs . Cambridge University Press, Cambridge, 2013. [Onthe cover: Selected topics in point-set topology]. doi:10.1017/CBO9781139524438 .[Gr¨a78] George Gr¨atzer.
General lattice theory , volume 75 of
Pure and Applied Mathematics .Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London,1978.[HL78] Karl H Hofmann and Jimmie D Lawson. The spectral theory of distributive continuouslattices.
Transactions of the American Mathematical Society , 246:285–310, 1978.[JS96] Achim Jung and Philipp S¨underhauf. On the duality of compact vs. open. In
Pa-pers on general topology and applications (Gorham, ME, 1995) , volume 806 of
Ann. New York Acad. Sci. , pages 214–230. New York Acad. Sci., New York, 1996. doi:10.1111/j.1749-6632.1996.tb49171.x .[Kei17] Klaus Keimel. The Cuntz semigroup and domain theory.
Soft Computing , 21(10):2485–2502, 2017.[PP12] Jorge Picado and Aleˇs Pultr.
Frames and locales: Topology without points .Frontiers in Mathematics. Birkh¨auser/Springer Basel AG, Basel, 2012. doi:10.1007/978-3-0348-0154-6 .[Pri70] H. A. Priestley. Representation of distributive lattices by means of ordered stonespaces.
Bull. London Math. Soc. , 2:186–190, 1970. doi:10.1112/blms/2.2.186 .[Shi52] Taira Shirota. A generalization of a theorem of I. Kaplansky.
Osaka Math. J. , 4:121–132, 1952. URL: http://projecteuclid.org/euclid.ojm/1200687806 .[Smy92] M. B. Smyth. Stable compactification. I.
J. London Math. Soc. (2) , 45(2):321–340,1992. doi:10.1112/jlms/s2-45.2.321 .[Sto36] M. H. Stone. The theory of representations for Boolean algebras.
Trans. Amer. Math.Soc. , 40(1):37–111, 1936. doi:10.2307/1989664 .[Sto38] M. H. Stone. Topological representations of distributive latticesand Brouwerian logics.
Cat. Mat. Fys. , 67(1):1–25, 1938. URL: http://hdl.handle.net/10338.dmlcz/124080 .[Vri62] H. De Vries. Compact spaces and compactifications: An algebraic approach. ThesisAmsterdam, 1962.[Wal38] Henry Wallman. Lattices and topological spaces.
Ann. of Math. (2) , 39(1):112–126,1938. doi:10.2307/1968717 . E-mail address ::