Towards a Gleason Cover for Compact Pospaces
aa r X i v : . [ m a t h . GN ] F e b Noname manuscript No. (will be inserted by the editor)
Towards a Gleason cover for compact pospaces
Laurent De Rudder · Georges Hansoul
Received: date / Accepted: date
Abstract
We establish a new category equivalent to compact pospaces, and whichextend the equivalence between compact Hausdorff spaces and Gleason spaces. Asa corollary of this equivalence, we obtain in particular, that every compact pospaceis the quotient of an f-space.
Keywords
Proximity · Compact pospaces · Duality theory · Gleason covers
Mathematics Subject Classification (2010) · · · · Introduction
The category
DeV of de Vries’ compingent algebras [10] can be considered amongneighbouring categories in a network of Stone-like and Gelfand-like dualities in-volving the categories
KHaus of compact Hausdorff spaces,
GlSp of Gleasonspaces,
KrFrm of compact regular frames and C ⋆ -alg of C ⋆ -algebras. KHaus DeV C ⋆ -alg GlSpKrFrm Laurent De RudderD´epartement de math´ematiques (B37)Universit´e de Li`egeBelgiumE-mail: [email protected] Laurent De Rudder, Georges Hansoul
In particular, the category
KHaus and
GlSp are equivalent, as established in[3]. This was first observed via the composition of the dualities between
KHaus and
DeV and between
DeV and
GlSp , then with a direct description.Categories of this base network were later generalized in different papers. In-deed, Bezhanishvili and Harding extended in [4] the dualities and equivalencesbetween
KHaus , KrFrm and
DeV to dualities and equivalence between the cat-egories
StKSp of stably compact spaces,
StKFrm of stably compact frames and
PrFrm of proximity frames. As for the duality between
KHaus and C ⋆ -alg , areal version of the duality, given in [5], was extended in [8] to a duality between KPSp of compact pospaces and the category usbal of Stone semirings. We referto [4] and [8] for the relevant definitions.
KHausKPSp DeV PrFrm C ⋆ -alg GlSpKrFrmStKFrm ?usbal The aim of this paper is to complete the extensions initiated in [4] and [8]to the category
GlSp . We point out that this extension process follows the samespirit as passing from Boolean algebras to distributive lattices, and from Stonespaces to Priestley spaces in the zero-dimensional setting (from the Boolean tothe distributive setting as we shall often say in this paper).The methodology goes as follows. First, we will establish on Priestley spacesthe counterpart of proximity relations on lattices. The road was well paved byCastro and Celani in [6], where the dual of a quasi-modal lattice (a generalizedproximity frame, but with a different class of morphisms) was already establishedas Priestley spaces endowed with an increasing closed binary relation. The obtainedtopological structures will be named ordered Gleason spaces and will be the objectsof a category whose morphisms are binary specific relations and not usual maps(as it is already the case in the Boolean setting [3]). Then, since duals of proximityframes in [4] were stably compact spaces, we will spend a few words on how todescribe them as compact pospaces. Finally, following Bezhanishvili steps in [2],we will show how to obtain directly the compact po-space dual to a Proximityframe via the latter’s Priestley dual. owards a Gleason cover for compact pospaces 3
In this section, we recall previous dualities which are essential for this paper,mainly for the sake of establishing notations that will be used throughout the restof the paper.
Priestley duality
We begin with the celebrated Priestley duality [14] and its characterization toframes in [15] through a suitable separation property.First of all, if ( X, ≤ , τ ) is an ordered topological space, we denote by τ ↑ (resp. τ ↓ ) the topology of open upsets (resp. open downsets) of τ .In particular, if ( X, ≤ , τ ) is a Priestley space, it is well known that τ ↑ (resp. τ ↓ )is generated by the clopen upsets (resp. clopen downsets) of X , which we denote byClop ↑ ( X ) (resp. Clop ↓ ( X )). Moreover, Clop ↑ ( X ) (or simply L , should the contextcause no confusion) is a distributive lattice when ordered by inclusion. Finally, if f : X −→ Y is an increasing continuous function between Priestley space, thenClop ↑ ( f ) : Clop ↑ ( Y ) −→ Clop ↑ ( X ) : O f − ( O )is a lattice morphism.On the other hand, if L is a bounded distributive lattice, we denote by Prim( L )(or more simply X ) its set of prime filters, ordered by inclusion and endowed bythe topology generated by { η ( a ) | a ∈ L } ∪ { η ( a ) c | a ∈ L } , where η ( a ) := { x ∈ Prim( L ) | x ∋ a } . Then Prim( L ) is a Priestley space and η is a lattice isomorphism between L andClop ↑ (Prim( L )). Moreover, if h : L −→ M is a lattice morphism thenPrim( h ) : Prim( M ) −→ Prim( L ) : x h − ( x )is an increasing continuous function. The functors Prim and Clop ↑ establish aduality between the categories DLat , of bounded distributive lattices, and
Priest ,of Priestley spaces.To continue, let us recall that a frame is a complete lattice L which satisfiesthe join infinite distributive law : for every subset S ⊆ L and every a ∈ L , we have a ∧ _ S = _ { a ∧ s | s ∈ S } . Furthermore, a lattice morphism h : L −→ M between two frames is a framemorphism if it preserves arbitrary joins. Lemma 1.1 ([15])
Let L be a frame and ( X, ≤ , τ ) be its Priestley dual.1. If O ∈ τ ↑ , then its closure in τ , denoted by cl( O ) , is an open upset.2. If S is a subset of Clop ↑ ( X ) , then W S = cl ( S { O | O ∈ S } ) .3. The map η : a η ( a ) is a frame morphism. Laurent De Rudder, Georges Hansoul
Refering to this result, an f-space is a Priestley space ( X, ≤ , τ ) which satisfiesthe first item of Lemma 1.1 and an increasing continuous function f : X −→ Y isan f -function if f − (cl( O )) = cl (cid:0) f − ( O ) (cid:1) for all O ∈ τ ↑ .Pultr and Sichler proved in [15] that Priestley duality reduces to a dualitybetween the categories Frm of frames and
FSp of f -spaces. Proximity frames
The second duality we recall was established by Bezhanishvili and Harding in [4],it can be seen as a generalization to frames and stably compact spaces (see [11,Definition VI-6.7.]) of de Vries duality.
Definition 1.2 A proximity frame is a pair ( L, ≺ ) where L is a frame and ≺ is a proximity relation , i.e. a binary relation on L such that – ≺ is a subordination relation S1. 0 ≺ ≺ a ≺ b, c implies a ≺ b ∧ c ,S3. a, b ≺ c implies a ∨ b ≺ c ,S4. a ≤ b ≺ c ≤ d implies a ≺ d , – which has the following additional propertiesS5. a = W { b ∈ L | b ≺ a } ,S6. a ≺ b implies a ≤ b ,S8. a ≺ b implies that a ≺ c ≺ b for some c .For the sake of convenience, we often identify the pair ( L, ≺ ) with its underlyingframe L .If S is a subset of L , we define ⇑ S := { b ∈ L | ∃ s ∈ S : s ≺ b } ( ⇓ S is defineddually). As usual, for an element a ∈ L , we write ⇑ a instead of ⇑{ a } . Definition 1.3 A proximity morphism is a map h : L −→ M between two prox-imity frames such that:H0. h is a strong meet-hemimorphism :(a) h (1) = 1 and h (0) = 0,(b) h ( a ∧ b ) = h ( a ) ∧ h ( b );H1. a ≺ b and a ≺ b implies h ( a ∨ a ) ≺ h ( b ) ∨ h ( b );H2. h ( a ) = W { h ( b ) | b ≺ a } .If h : L −→ M and g : M −→ N are proximity morphisms, their composition isdefined by g ⋆ h : L −→ N : a _ { g ( h ( b )) | b ≺ a } . We denote by
PrFrm the category of proximity frames endowed with proximitymorphisms.
Definition 1.4 If L is a proximity frame, a round filter of L is a lattice filter F such that F = ⇑ F . We denote by RF ( L ) the set of all round filters of L .An end is a round filter p such that for every round filters F , F , we have F ∩ F ⊆ p if and only if F ⊆ p or F ⊆ p . We denote by End( L ) (or only by P )the set of all ends of L . The seemingly peculiar way used to denote the properties of ≺ (and the absence of S7)stems from the works on subordination and (pre-)contact algebras, see for instance [3], [9] or[13].owards a Gleason cover for compact pospaces 5 The ends of Definition 1.4 will now play a role similar to the one of prime filtersin Priestley duality. Indeed, endowed with the topology generated by the sets ofthe form µ ( a ) := { p ∈ End( L ) | p ∋ a } , (1)End( L ) is a stably compact space. Moreover, if h : L −→ M is a proximity frame,then End( h ) : End( M ) −→ End( L ) : p h − ( p )is a proper continuous function.On the topological side, if ( P, τ ) is a stably compact space, then τ := Ω ( P )ordered by inclusion is a proximity frame when endowed with the relation ≺ definedby O ≺ V if and only if O ⊆ K ⊆ V for some compact subset K . Furthermore, if f : P −→ Q is a proper continuous function between two stably compact spaces,then Ω ( f ) : Ω ( Q ) −→ Ω ( P ) : O f − ( O )is a proximity morphism.Now, the functors End and Ω establish a duality between PrFrm and thecategory
StKSp of stably compact spaces (see [4, Theorem 4.18]).
In addition to the duality between
PrFrm and
StKSp , we can provide a modal-like duality between
PrFrm and a category of f-spaces endowed with a particularbinary relation R . Following the taxonomy of [2], we name ordered Gleason spaces the pairs f -spaces/relations obtained. At the objects level, we can rely on theworks previously done in [6] for quasi-modal lattices and in [3] for the Booleansetting. Hence, most of the proofs are left to the reader. Definition 2.1 An ordered Gleason space is a triple ( X, ≤ , R ) where ( X, ≤ ) is an f -space and R is a binary relation on X satisfying the following properties:1. R is closed in X ;2. x ≤ y R z ≤ t implies x R t ;3. R is a pre-order;4. For every O ∈ Clop ↑ ( X ), we have O = cl ( R [ − , O c ] c ).An equivalent definition is given by substituting 2 with2’. x ≤ y implies x R y . Remark 2.2
Let us highlight some observations and introduce notations that wefreely use in the rest of the paper. – Let R be a binary relation on an arbitrary set X :1. If E is a subset of X , we note R [ − , E ] := { x | ∃ y ∈ E : x R y } and R [ E, − ] := { x | ∃ y ∈ E : y R x } . For an element x ∈ X , we note R [ − , x ] instead of R [ − , { x } ]. Note that, if( L, ≺ ) is a proximity frame, then we have ⇑ x = ≺ [ x, − ]. The relations between proximity/subordination relations and quasi-modal operator is welldiscussed for instance in [7]. Laurent De Rudder, Georges Hansoul
2. If E and F are subsets of X , then R [ − , E ] ⊆ F if and only if R [ F c , − ] ⊆ E c .
3. If X is a topological space, R is closed in X and F is a closed subset of X , then R [ − , F ] and R [ F, − ] are closed. – Let L be a distributive lattice and S an arbitrary subset of L . We define F S := { x ∈ Prim( L ) | S ⊆ x } . Remark that F S = T { η ( a ) | a ∈ S } so that F S is a closed (and hence acompact) subset of Prim( L ).The future duality between proximity frames and ordered Gleason spaces isnow obtained as follows. Let ( L, ≺ ) be a proximity frame, its dual is given by( X, R ) where X = Prim( L ) is the Priestley dual of L and R is the binary relationon X defined by x R y if and only if ⇑ x ⊆ y. (2)Let us highlight the fact that equivalent definitions of the relation R are given by ⇑ x ⊆ ⇑ y or ⇓ y c ⊆ ⇓ x c or ⇓ y c ⊆ x c . (3) Lemma 2.3
Endowed with the relation R defined in (2) , Prim( L ) is an orderedGleason space. Furthermore, for every a, b ∈ L , we have a ≺ b if and only if R [ η ( a ) , − ] ⊆ η ( b ) . Proof (Sketch of the proof)
To prove Items 1 and 2 of Definition 2.1, one just hasto use the subordination part of a proximity relation (see Definition 1.2). Also,one can show that R is reflexive if and only if ≺ satisfies S6 and transitive if andonly ≺ satisfies S8. Let us prove item 5 (which is equivalent to S5.). We have a = W { b | b ≺ a } if and only if η ( a ) = η ( W { b | b ≺ a } ). Then, by [15, Theorem1.5], it comes that η (cid:16)_ { b | b ≺ a } (cid:17) = cl (cid:16)[ { η ( b ) | b ≺ a } (cid:17) = cl (cid:16)[ { η ( b ) | R [ η ( b ) , − ] ⊆ η ( a ) } (cid:17) = cl (cid:16)[ { η ( b ) | η ( b ) ⊆ R [ − , η ( a ) c ] c } (cid:17) . Finally, since R [ − , η ( a ) c ] c is an open upset , it follows that η (cid:16)_ { b | b ≺ a } (cid:17) = cl ( R [ − , η ( a ) c ] c ) , and the conclusion is clear.On the other hand, let ( X, ≤ , R ) be an ordered Gleason space, its dual is givenby ( L, ≺ ) where L := Clop ↑ ( X ) is the Priestley dual of X and ≺ is the binaryrelation on L defined by O ≺ U if and only if R [ O, − ] ⊆ U. (4) owards a Gleason cover for compact pospaces 7 Lemma 2.4
Endowed with the relation ≺ defined in (4) , Clop ↑ ( X ) is a proximityframe. To conclude the section, it remains to determine the counterpart of the prox-imity morphisms on Gleason spaces. Let L and M be proximity frames and X and Y their respective Priestley duals. If h : L −→ M is a meet-hemimorphism, thenthe relation ρ ⊆ Y × X defined by y R x if and only if h − ( y ) ⊆ x (5)satisfy the following conditions:1. y ≤ y ρ x ≤ x implies y ρ x ,2. ρ is closed in Y × X ,3. O ∈ Clop ↑ ( X ) implies ρ [ − , O c ] c ∈ Clop ↑ ( Y ).Since in our case, we have strong meet-hemimorphism, the relation ρ alsosatisfies4. for every y ∈ Y , there exists x ∈ ρ [ y, − ].We call such a relation ρ a strong meet-hemirelation . By [17, Lemma 2], weknow that strong meet-hemimorphism are in correspondence with strong meet-hemirelation. Hence, it remains to characterize the properties H1 and H2 of prox-imity morphisms. A key concept towards this characterization is defined below. Definition 2.5
Let ( X, ≤ , R ) be an ordered Gleason space and S a subset of X .An element x ∈ S is said to be R -minimal in S if for every y ∈ S , y R x implies x R y . Proposition 2.6
Let ( X, R ) be an ordered Gleason space and F be a closed subsetof X , then for every element x ∈ F there exists an element y R -minimal in F suchthat y R x .Proof We follow the lines of the proof for po-sets (see for instance [11, PropositionVI.5-3.]) Let us define a chain of (
X, R ) to be a subset C of X such that for every x, y ∈ C , we have x R y or y R x .We denote by C the set of chains C satisfying x ∈ C ⊆ F , ordered by inclusion.We have that C is non-empty (by reflexivity of R , it contains the chain { x } ) and aclassical argument suffices to prove it is also inductive. Hence, C admits a maximalelement M .Since { R [ − , z ] ∩ F | z ∈ M } is a family of closed sets which satisfies the finiteintersection properties (because M is a chain contained in F and R is a pre-order),we know by compactness that there exists an element y ∈ F such that y R z forall z ∈ M .Now, suppose that t is an element of F such that t R y . By transitivity, wehave that { t } ∪ M is a chain of C . By maximality of M , we have t ∈ M and,therefore, we have y R t , so that y is indeed R -minimal in F , as required.Let us highlight that the notion of R -minimal element is also present in theBoolean setting, while hidden. Indeed, in the Boolean case, the relation R turnsout to be an equivalence relation, so that every element is actually R -minimal. Laurent De Rudder, Georges Hansoul
Proposition 2.7
Let h : L −→ M be a strong meet-hemimorphism between twoproximity frame and ρ ⊆ Y × X its associated strong hemi-relation:1. h satisfies H1 if and only if for every y , y ∈ Y , every x R-minimal in ρ [ y , − ] and every x ∈ X , we have x ρ − y R y ρ x implies x R x . h satisfies H2 if and only if ρ [ − , O c ] = int ( ρ [ − , R [ − , O c ]]) for every O ∈ Clop ↑ ( X ) . The proof of Item 2 is almost identical to the one in the Boolean case. Therefore,we redirect the reader to [3, Lemma 6.11] for more details. The proof of Item1 requires additional results. In the meantime, we name ordered forth condition (shortened as ofc ) and de Vries condition (shortened as dvc ) the conditions of thefirst and the second item of Proposition 2.7.Before we start, let us note that ρ [ − , η ( a ) c ] c = η ( h ( a )) . (6)Indeed, it is clear that η ( h ( a )) ⊆ ρ [ − , η ( a ) c ] c . Now, suppose that y ∈ ρ [ − , η ( a ) c ] c .Then, for every x ∈ η ( a ) c , we have that h − ( y ) x . Hence, for all x ∈ η ( a ) c ,we have that h ( a x ) ∈ y and a x x for some a x ∈ L . In particular, { η ( a x ) c | x ∈ η ( a ) } is an open cover of η ( a ) c which is compact. Then, we now that there exist x , . . . , x n ∈ η ( a ) c such that η ( a x ∧ · · · ∧ a x n ) c ⊇ η ( a ) c . Moreover, we have ( y is a filter) y ∋ h ( a x ) ∧ · · · ∧ h ( a x n ) = h ( a x ∧ · · · ∧ a x n ) ≥ h ( a )and the conclusion is clear. Proposition 2.8
Let
L, M be two proximity frames, h : L −→ M be a proximitymorphism , y a prime filter of M and x a prime filter which is R -minimal in ρ [ y, − ] . Then, we have ⇑ (cid:16) h − ( ⇑ y ) (cid:17) = ⇑ x. Proof
On the one hand, h − ( ⇑ y ) ⊆ x follows from x ∈ ρ [ y, − ]. Consequently, wehave ⇑ (cid:0) h − ( ⇑ y ) (cid:1) ⊆ ⇑ x .On the other hand, suppose that ⇑ x
6⊆ ⇑ (cid:0) h − ( ⇑ y ) (cid:1) . Then, there exist a ∈ x and b ∈ L such that a ≺ b and b
6∈ ⇑ (cid:0) h − ( ⇑ y ) (cid:1) . By the properties of proximityrelations, we know that a ≺ c ≺ d ≺ e ≺ b for some c , d , e ∈ L . Inparticular, we have h ( d ) ≺ h ( e ) and e h − ( ⇑ y ). Therefore, we also have that h ( d ) y .In order to obtain an absurdity and conclude the proof, we are going to inval-idate the R -minimality of x in ρ [ y, − ].We first prove that h − ( y ) ∩ h c ∪ ⇓ x c i id = ∅ , (7) owards a Gleason cover for compact pospaces 9 where h c ∪ ⇓ x c i id is the lattice ideal generated by c ∪ ⇓ x c . Suppose this is notthe case. Then, there exist a , c ∈ L and d ∈ x c such that h ( a ) ∈ y , c ≺ d and a ≤ c ∨ c . It follows from the properties of h that y ∋ h ( a ) ≤ h ( c ∨ c ) ≺ h ( d ) ∨ h ( d ) . Now, since y is a prime filter and h ( d ) y , we have that h ( d ) ∈ y . Hence, wehave d ∈ h − ( y ) ⊆ x , which is absurd. Consequently, (7) is satisfied and we have h − ( y ) ⊆ z , c z and ⇑ z ⊆ x for some prime filter z . In other words, we have z R x and y ρ z . Now, by R -minimality of x in ρ [ y, − ], it follows that x R z .Hence, in particular, we should have c ∈ ⇑ a ⊆ ⇑ x ⊆ z, which is absurd.We now have the required result to finish the proof of Proposition 2.7. Proof (Proof of Proposition 2.7)
For the only if part, suppose that h − ( y ) ⊆ x , h − ( y ) ⊆ x and ⇑ y ⊆ y . In particular, by Proposition 2.8, we have ⇑ (cid:0) h − ( ⇑ y ) (cid:1) = ⇑ x . It comes that ⇑ x = ⇑ (cid:16) h − ( ⇑ y ) (cid:17) ⊆ h − ( ⇑ y ) ⊆ h − ( y ) ⊆ x , or, in other words, that x R x , as required.For the if part, let a , a , b and b be elements of L such that a ≺ b and a ≺ b . To prove that h satisfies H1 is to prove that R [ η ( h ( a ∨ a )) , − ] ⊆ η ( h ( b )) ∪ η ( h ( b )) . We can use (6) to rewrite this inclusion as R [ ρ [ − , η ( a ∨ a ) c ] c , − ] | {z } := A ⊆ ρ [ − , η ( b ) c ] c ∪ ρ [ − , η ( b ) c ] c | {z } := B . Let y ∈ A . Then, there exists y such that y R y and such that ρ [ y , − ] ⊆ η ( a ∨ a ). Moreover, by Proposition 2.6, we know that there exists a filter x R -minimal in ρ [ y , − ]. Hence, we may suppose, without loss of generality, that a ∈ x . Let x be a prime filter such that y ρ x . By the ofc, we know that x R x and it follows that b ∈ ⇑ a ⊆ ⇑ x ⊆ x . Hence, we proved that for every x such that y ρ x , we have x ∈ η ( b ), that is y ∈ ρ [ − , η ( b ) c ] c ⊆ B , as required.Now that we characterized the strong meet-hemirelation that stemmed fromproximity morphisms, we have to determine how to compose them to actuallyobtain category dual to PrFrm . As it was already noted in [3], the rule of com-position of meet-hemirelations is not easily described, even in the Boolean settingand we must rely on their associated meet-hemimorphisms.
Definition 2.9
Let ρ and ρ be meet-hemirelations and h , h their associatedmeet-hemimorphisms. We define the composition ρ ⋆ ρ as the meet-hemirelationassociated to h ⋆ h .With all of the above observations, the next definition and theorem come asno surprise. Definition 2.10
We denote by
OGlSp the category whose objects are orderedGleason spaces and whose morphisms are strong meet-hemirelations which satisfythe ofc and the dvc, with the composition of Definition 2.9. For the record, let usnote that the identity morphisms in
OGlSp are given by the order relations ofthe ordered Gleason spaces.
Theorem 2.11
The categories
OGlSp and
PrFrm are dual to each other.
Of course, as direct corollary of Theorem 2.11 and [4], the categories
OGlSp and
StKSp are equivalent. The scope of the next section is to describe directly thisequivalence.. However, since this paper is ”ordered-minded”, we swap the category
StKSp for its equivalent category
KPSp of compact pospaces, also sometimescalled Nachbin spaces. A compact pospace is a triple ( P, π, ≤ ) where ( P, π ) is a compactspace and ≤ is an order relation on P which is closed in P . We denote by KPSp the category of compact pospaces and continuous monotone maps.The equivalence between
KPSp and
StKSp is almost folklore (see for instance[11, Section VI-6]). We recall here the basic facts.If (
P, τ ) is a stably compact space, then (
P, π, ≤ τ ) is a compact pospace where π is the patch topology associated to τ and ≤ τ is the canonical order on ( P, τ ),that is p ≤ τ q if and only if p ∈ cl τ ( { q } ). In addition, we have π ↑ = τ and π ↓ is the co-compact topology associated to τ , that is the compact saturated sets of τ . On the other hand, if ( P, π, ≤ ) is a compact pospace, then ( P, π ↑ ) is a stablycompact space. With these consideration in mind, we can describe the ends spaceEnd( L ) of a proximity frame L as a compact pospace. Proposition 3.2
Let L be a proximity frame and P := End( L ) its ends space.1. For p, q ∈ P , p ≤ q if and only if p ⊆ q ,2. The topology π ↑ is generated by the sets µ ( a ) for a ∈ L (see (1) ),3. The closed elements of π ↓ are given by the sets of the form K F := { p ∈ P | p ⊆ F } ; for some round filter F .Proof Item 2 is immediate. For item 1, we have p ≤ q ⇐⇒ p ∈ cl τ ( { q } ) = ∩{ µ ( a ) c | q ∈ µ ( a ) c }⇐⇒ ∀ a ∈ L : a q ⇒ a p ⇐⇒ q c ⊆ p c ⇐⇒ p ⊆ q. To prove Item 3, we use several results established in [4]. owards a Gleason cover for compact pospaces 11
1. From [4, Lemma 4.14], there is a homeomorphism α from End( L ) to pt RI ( L ) given by α : pt( RI ( L )) −→ End( L ) : g p g := { a ∈ L | g ( ⇓ a ) = 1 } .
2. From [4, Remark 4.21], there is a bijection between RF ( L ) and the Scott-openfilters of RI ( L ) given by F I ∈ RI ( L ) | F ∩ I = ∅} .
3. From [11, Theorem II.1.20], there is an order-reversing bijection between Scott-open filter of RI ( L ) and the the compact saturated sets of pt( RI ( L )) whichis given by F p ∈ pt( RI ( L )) | ∀ I ∈ F : g ( I ) = 1 } . Consequently, there is a bijection between the compact saturated sets of pt( RI ( L ))and RF ( L ) which is given by F g ∈ pt( RI ( L )) | ∀ I ∈ RI ( L ) : I ∩ F = ∅ ⇒ g ( I ) = 1 } . Then, since α is a homeomorphism, we now that there is a bijection between thecompact saturated sets of End( L ) and RF ( L ) given by F p ∈ End( L ) | ∀ I ∈ RI ( L ) : I ∩ F = ∅ ⇒ I ∩ p = ∅ | {z } = ⋆ } . Finally, to conclude the proof, let us show that the condition ⋆ is equivalent to F ⊆ p . Clearly, if F ⊆ p , then ⋆ is satisfied. Now, suppose that F p . Then, thereexists an element a ∈ F \ p . In particular, we have that ⇓ a is a round ideal suchthat ⇓ a ∩ p = ∅ and, since F is round, such that ⇓ a ∩ F = ∅ such that the condition ⋆ is not satisfied.Now, we focus on how End( L ) relates with Prim( L ). A first immediate remarkis that for every round filter F and every prime filter x , we have F ⊆ x ⇔ F ⊆ ⇑ x. (8)A second step is undertaken in the following lemma. Lemma 3.3
Let L be a proximity frame. For every prime filter x ∈ Prim( L ) , ⇑ x is an end of L .Proof It is clear that ⇑ x is a filter but, by S8 of Definition 1.2, it is also a roundfilter. Moreover, if F and F are round filters such that F ∩ F ⊆ ⇑ x . In par-ticular, this implies that F ∩ F ⊆ x . Now, x being a prime filter, we know that F ⊆ x or F ⊆ x . It then follows from (8) that ⇑ x is an end.Our goal now is to prove that every end is of the form ⇑ x for some prime filter x . We start with the next proposition. The points of a frame M are frame morphisms g from M into , the two elements frame.We denote by pt( M ) the set of all points of M . A round ideal of a proximity frame is a latticeideal I such that ⇓ I = I . The set RI ( L ) of all round ideals of L is a frame when order byinclusion.2 Laurent De Rudder, Georges Hansoul Proposition 3.4
Let L be a proximity frame. There is a bijection between theround filters of L and the R -increasing closed subsets of X = Prim( L ) , given by Φ : F F F := { x ∈ X | x ⊇ F } . (9) Proof
First, it is clear that F F is a closed set and that it is R -increasing, so that Φ is well defined. Moreover, Φ is one-to-one since every filter is the intersectionof the prime filters containing it. Finally, we show that Φ is onto. Let F be an R -increasing closed set. In particular, F is an increasing closed subset (recallDefinition 2.1), and, therefore, we know that F = \ { η ( a ) | η ( a ) ⊇ F } . If we set F = { a | η ( a ) ⊇ F } , then x ∈ F F ⇔ F ⊆ x ⇔ ( η ( a ) ⊇ F ⇒ x ∈ a ) ⇔ x ∈ F. Since one can show that F is a filter by routine calculations, it remains to prove itis round.Let a be an element of F . As F is an R -increasing set, it comes that R [ F, − ] ⊆ η ( a ). Recall that R is a closed relation and that { η ( b ) | η ( b ) ⊇ F } is a filteredfamily of closed sets such that F = T { η ( b ) | η ( b ) ⊇ F } . Hence, by Esakia Lemma(see for instance [16, p. 995]), it follows that η ( a ) ⊇ R [ F, − ] = R [ \ η ( b ) , − ] = \ R [ η ( b ) , − ] . It is now sufficient to use compactness to obtain R [ η ( b ) ∩ · · · ∩ η ( b n ) , − ] ⊆ R [ η ( b ) , − ] ∩ · · · ∩ R ([ η ( b n ) , − ] ⊆ η ( a )for some b , . . . , b n . If we set b := b ∧ · · · ∧ b n , we have F ⊆ η ( b ) and R [ η ( b ) , − ] ⊆ η ( a ) , that is b ∈ F and b ≺ a as required.Let us note that the application Φ defined in (9) is a reverse order isomorphism,in the sense that for two round filters F and F ′ , we have F ⊆ F ′ if and only Φ ( F ) ⊇ Φ ( F ′ ). Therefore, the R -increasing closed sets which are associated to endsare exactly the join-prime R -increasing closed sets. We will use this observationand the next definition to prove the reciprocal of Lemma 3.3. Definition 3.5
Let ( X, ≤ , R ) be an ordered Gleason space. We denote by ≡ theequivalence relation associated to the pre-order R , i.e. x ≡ y if and only if x R y and y R x. Since R is closed, ≡ is also closed. Moreover, X/ ≡ ordered by x ≡ ≤ R y ≡ if and only if x R y for the order of X owards a Gleason cover for compact pospaces 13 is a compact pospace.We highlight the fact that, if ( X, ≤ , R ) is the dual of a proximity frame ( L, ≺ ),then the equivalence relation ≡ can be expressed as follow: x ≡ y if and only if ⇑ x = ⇑ y, or, equivalently, x ≡ y if and only if ⇓ x c = ⇓ y c . Lemma 3.6
Let ( L, ≺ ) be a proximity frame. If p ∈ End( L ) , then there exists aunique ≡ -class x ≡ such that x is R-minimal x and F p = R [ x, − ] .Proof We know that for every element z ∈ F p , there exists an R -minimal element x ∈ F p such that x R z . Hence, it remains to prove its uniqueness.Suppose that there exist two R -minimal elements x and y in F p such that x y . In other words, we have x R y and x R y . Using a classical argument, onecan show that there exist two R -decreasing open sets ω and ω such that x ∈ ω , y ∈ ω and ω ∩ ω = ∅ . In other words, such that F p = F p \ ω ∪ F p \ ω . Since F p \ ω i are R -decreasing closed sets and F p is join-prime (recall the discussion afterProposition 3.4), it follows that F p ⊆ F p \ ω or F p ⊆ F p \ ω , which is of courseimpossible since, for instance, x ∈ F p and x ∈ ω . Theorem 3.7
Let ( L, ≺ ) be a proximity frame. A subset p ⊆ L is an end if andonly if p = ⇑ x for some x ∈ Prim( L ) .Proof The if part is Lemma 3.3. For the only part, let p be an end. By Lemma3.6, we have F p = R [ x, − ] for some prime filter x . In particular, it follows that Φ ( p ) = Φ ( ⇑ x ) and therefore that p = ⇑ x , as required.It follows from Theorem 3.7 that, at least for the underlying sets, End( L ) isthe quotient of Prim( L ) by the relation ≡ . We denote by σ the application σ : Prim( L ) / ≡ −→ End( L ) : x ≡ x. We now want to prove that End( L ) is the quotient of Prim( L ) as ordered topo-logical spaces, that is prove that σ is an order homeomorphism. Theorem 3.8
Let ( L, ≺ ) be a proximity frame. Then, in the category KPSp , wehave
End( L ) ∼ = Prim( L ) / ≡ , by the application σ .Proof First, we have that σ is onto by Theorem 3.7 and that it is one-to-one bydefinition. We also have that σ is an order isomorphism, since we have x ≡ ≤ R y ≡ ⇔ x R y ⇔ ⇑ x ⊆ ⇑ y ⇔ σ ( x ≡ ) ≤ σ ( y ≡ ) . Therefore, since End( L ) and Prim( L ) / ≡ are compact Hausdorff spaces, it remainsto prove σ is continuous. By Proposition 3.2, we have to prove that σ − ( µ ( a )) and σ − ( K F ) are respectively open and closed for every a ∈ L and F ∈ RI ( L ). Let Π : x x ≡ be the canonical quotient map. We have: Π − ( σ − ( µ ( a ))) = { x ∈ Prim( L ) | a ∈ ⇑ x } = [ { η ( b ) | b ≺ a } which is open, and Π − ( σ − ( K F )) = { x ∈ Prim( L ) | F ⊆ x } = F F , which is closed.With Theorem 3.8, we can describe a functor from the category OGlSp to thecategory
KPSp which sends an ordered Gleason space ( X, ≤ , R ) to the compactpospace ( X/ ≡ , R ). Proposition 2.8 gives a hint on how to deal with the morphisms.Indeed, if ρ ⊆ X × Y is a strong meet-hemirelation which satisfies the ofc and thedvc between ordered Gleason spaces, then we know that it can be associated witha proximity morphism h : Clop ↑ ( Y ) −→ Clop ↑ ( X ) : O ρ [ − , O c ] c . Then, thismorphism is associated with the continuous function f : End(Clop ↑ ( X )) −→ End(Clop ↑ ( Y )) : p h − ( p ) . Now, p is equal to ⇑ x for some x ∈ X and if y is an R -minimal element in ρ [ x, − ],that is such that h − ( x ) ⊆ y , we have f ( p ) = f ( ⇑ x ) = ⇑ h − ( ⇑ x ) = ⇑ y. Hence, we have to send a meet hemirelation ρ to the function f ρ : X/ ≡ −→ Y / ≡ : x ≡ y ≡ for y R -minimal ρ [ x, − ] . By the dualities between
KPSp and
PrFrm and between
PrFrm and
OGlSp , f is an increasing continuous function. But, we have a direct proof. Proposition 3.9
Let ρ ⊆ X × Y be a strong hemirelation that satisfies ofc anddvc between two ordered Gleason spaces. The map defined by f : X/ ≡ −→ Y / ≡ : x ≡ y ≡ for y R-minimal in ρ [ x, − ] is an increasing continuous function.Proof First, the ofc implies that f is well defined and increasing. Now, since Y / ≡ is a compact pospace, to prove that f is continuous, it is enough to prove that f − ( ω ) and f − ( F ) are respectively open and closed subsets of X/ ≡ for ω an opendownset and F a closed downset of Y / ≡ .For ω , we have x ≡ ∈ f − ( ω ) ⇐⇒ ∃ y ∈ Π − ( ω ) : y is R -minimal in ρ [ x, − ] (10) ⇐⇒ x ∈ ρ [ − , Π − ( ω )] (11)While the implication (10) ⇒ (11) does not need to be proved, a word must bespent on the reciprocal. Suppose that x ∈ ρ [ − , π − ( ω )], then we have x ρ y forsome z ∈ Π − ( ω ). By Proposition 2.6, there exists y R-minimal in ρ [ x, − ] suchthat y R z . Now, ω is a downset of Y / ≡ , so that Π − ( ω ) is an R -decreasing subsetof Y . It follows that y ∈ Π − ( ω ), as required. Restarting from (11), since Π − ( ω )is R -decreasing and open, it is in particular an open downset of Y . Therefore, Π − ( ω ) = [ { O ∈ Clop ↓ ( Y ) | O ⊆ Π − ( ω ) } owards a Gleason cover for compact pospaces 15 and, consequently, ρ [ − , Π − ( ω )] = [ { ρ [ − , O ] | O ⊆ Π − ( ω ) } . By the dvc (see Proposition 2.7), ρ [ − , O ] is an open subset of X for every O ∈ Clop ↓ ( Y ) and, hence, so is ρ [ − , Π − ( ω )]. Henceforth, we proved that Π − ( f − ( ω )) = ρ [ − , Π − ( ω )] is open in X , as required.Finally, as for ω , we have that Π − ( f − ( F )) = ρ [ − , Π − ( F ) . ]Now, since ρ is closed in X × Y , ρ [ − , Π − ( F )] is a closed subset of X and theproof is concluded.Hence, we have a functor ξ between the categories OGlSp and
KPSp whichmaps an ordered Gleason space ( X, ≤ , R ) to the compact pospace ( X/ ≡ , ≤ R ) anan ordered Gleason relation ρ ⊆ X × Y to the increasing continuous function f : X/ ≡ −→ Y / ≡ defined in Proposition 3.9. This functor yields an equivalencebetween OGlSp and
KPSp which is equivalent to the composition of the dualitybetween
OGlSp and
PrFrm and the duality between
PrFrm and
KPSp . Remark 3.10
In the Boolean setting, an important feature of Gleason spacesis that their underlying stone spaces are the projective objects in the category
KHaus . This is not the case anymore in our distributive setting. Indeed, the f -spaces are not the projective objects of the category KPSp since it would impliesthat they are projective in the category
Priest . However, the injective objects of
DLat have been shown in [1] to exactly be the complete Boolean algebras andnot the frames. In fact, the projective objects of
KPSp are exactly the projectiveobjects of
KHaus , that is the extremally disconnected compact spaces, as weshow in the next proposition.
Proposition 3.11
The projective objects in the category
KPSp are exactly theextremally disconnected spaces (ordered by equality).Proof
First, let us consider ( X, =) a extremally disconnected space, ( P, ≤ ) and( Q, ≤ ) compact po-spaces, f : X −→ P a monotone continuous function and g : Q −→ P a surjective monotone continuous function. Since every compact po-space is in particular compact Hausdorff, and since the extremally disconnectedspaces are projective in KHaus , there exists a continuous function h : X −→ Q such that gh = f and, since X is ordered by the equality, h is clearly monotone.Hence, ( X, =) is indeed projective in KPSp .On the other hand, suppose that (
X, τ, ≤ ) is projective in KPSp . Then, fol-lowing the proof of Gleason in [12], one can prove that (
X, τ ) is extremally dis-connected.The main “ethical” reason behind the failure of ordered Gleason spaces asprojective objects in
KPSp is the the relation R is submerged by its associatedequivalence relation ≡ . A solution could be to change the properties of the mor-phisms in the projective problem so that they directly take into account R insteadof ≡ . Conclusion
We have completed the external network of equivalences and dualities started in[4] and [8], generalising to the ”distributive setting” the duality between Gleasonspaces and compact Hausdorff spaces of [3]. Hence, we obtain the following commu-tative diagram, where the arrowed lines represent adjunctions and the non-arrowedones equivalences or dualities.
DeVPrFrm KHausKPSp C ⋆ -alg GlSpKrFrmStKFrm OGlSpusbal However, A proper way to describe the functor between
KPSp and
OGlSp is stillmissing. This situation could be solved figuring out the universal problem answeredby ordered Gleason spaces. This problem cannot be the usual projective one as wesaw at the end of Section 3. We will address this problem in a forthcoming article.
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