NOTE ON PRIESTLEY-STYLE FOR DISTRIBUTIVE MEET-

MAR´IA ESTEBAN

Abstract. We carry out a detailed comparison of the two topological dualities for distributive meet-semilattices studied by Celani [3] and by Bezhanishvili and Jansana [2]. We carry out such comparison, that was already sketched in [2], by defining the involved in the equivalence of both dual categories of distributive meet-semilattices.

1. Introduction Distributive meet-semilattices are ordered algebraic structures that play a promi- nent role in the algebraic study of logic (MV-algebras, residuated lattices, etc). Celani studied in [3] a duality for distributive meet-semilattices with top element, that can be easily restricted to a duality for bounded distributive meet-semilattices. In such duality, prime (i. e. irreducible) filters are used to represent and dually char- acterize distributive meet-semiattices. Bezhanishvili and Jansana studied in [2] a different duality for bounded distributive meet-semilattices, that is based on a di- fferent collection of filters, namely optimal filters. In [2] such duality is named as a Priestley-style duality, whereas Celani’s duality is named as a Spectral-like duality. It is remarkable that, unlike the case of distributive lattices, in which Spectral and Priestley dualities are both built on the collection of prime filters, the aforesaid dualities for distributive meet-semilattices are built on different collections of filters. However, the two dualities can be seen as generalizations of the respective dualities for distributive lattices. In the last section of [2], a brief comparison between the referred dualities for (bounded) distributive meet-semilattices is carried out, and it is shown how to move from one to the other, but not the opposite sense. The purpose of this note is to explicitly define the functors that are obtained as the concatenation of the two dual constructions, passing through the algebraic one, and that witness that the two categories dual to bounded distributive meet-semilattices are in fact equivalent.

2. Dualities for bounded distributive meet-semilattices An algebra M = hM, ∧, 1, 0i of type (2, 0, 0) is a bounded meet- when the binary operation ∧ is idempotent, commutative, associative, and a ∧ 1 = a and a ∧ 0 = 0 for all a ∈ M. A partial order is defined in M by a ≤ b if and only if a ∧ b = a. A bounded meet-semilattice M is distributive (cf. Section II.5 in [5]) when for each a, b1, b2 ∈ M with b1 ∧ b2 ≤ a, there exist c1, c2 ∈ M such that b1 ≤ c1, b2 ≤ c2 and a = c1 ∧ c2.

Date: December 13, 2013. 1 2 MAR´IA ESTEBAN

We address the reader to [2] and [4] for the main definitions of filters and ideals of distributive meet-semilattices, such as order ideals (non-empty up-directed down- sets), Frink ideals (down-sets closed under existing finite joins), order filters (non- empty down-directed up-sets), irreducible filters (meet-irreducible elements of the of order filters) and optimal filters (order filters that are complements of Frink ideals)1. For any bounded distributive meet-semilattice M, we denote such collections by Id(M), IdF (M), Fi(M), Pr(M) and Op(M) respectively. We review now the dualities in [2] and [3] for the of bounded distributive meet-semilattices and algebraic homomorphisms. We denote such category by BDS and we follow hereinafter the notation and terminology used in [2]. In particular, for the composition of relations we use the standard set-theoretic notation. Recall that a Priestley space is an ordered hX, τ, ≤i that is compact and totally order-disconnected. For a dense subset X0 of X, and a clopen subset U of X, we say that U is X0-admissible provided max(X \ U) ⊆ X0. The duality for BDS studied in [2] by Bezhanishvili and Jansana can be described as a Priestley-style duality, since dual objects are Priestley spaces equipped with additional structre. Regarding objects, their strategy consists of what follows: first they study how any bounded distributive meet-semilattice M can be embedded in a bounded L(M), that they call the distributive envelope of M. Thus they associate as the Priestley dual of M, the Priestley dual of L(M), that can be described in terms of optimal filters of M. But they need to work with additional structure in order to recover the original semilattice.

Definition 2.1. (Definition 5.5 in [2]) A quadruple X = hX, τ, ≤,X0i is a gener- alized Priestley space when: (Pr1) hX, τ, ≤i is a Priestley space, (Pr2) X0 is a dense subset of X, (Pr3) for each x ∈ X there is y ∈ X0 such that x ≤ y, ∗ (Pr4) x ∈ X0 iff {U ∈ X : x∈ / U} is up-directed, (Pr5) for all x, y ∈ X we have x ≤ y iff (∀U ∈ X∗) if x ∈ U, then y ∈ U.

∗ Recall that X denotes the collection of all X0-admissible clopen up-sets of X. From the definition it follows that X∗ ∪ {X \ U : U ∈ X∗} is a subbasis for τ. Moreover X∗ := hX∗, ∩, ∅,Xi is a bounded distributive meet-semilattice, called the dual semilattice of X.

Definition 2.2. (Definition 6.2 in [2]) Let X and Y be generalized Priestley spaces. A R ⊆ X × Y is called a generalized Priestley morphism when: (PrR1) If (x, y) ∈/ R, then there is U ∈ Y ∗ such that y∈ / U and R[x] ⊆ U, where R[x] := {y ∈ Y :(x, y) ∈ R}, ∗ ∗2 (PrR2) if U ∈ Y , then R(U) ∈ X .

1Note that the definitions of ‘Frink ’ and ‘optimal filter’ provided in [2] are appropriate only for distributive meet-semilattices that are bounded. When there is no necessarly a bottom element, we need to slightly modify such definitions, and to adopt instead the ones provided in [4] (see p. 28 for a discussion on this issue). 2 Where R : P(Y ) −→ P(X) is defined for any sets X,Y and any relation R ⊆ X × Y by: R(A) = {x ∈ X : R[x] ⊆ A}. Notice that for sets X,Y,Z and relations R ⊆ X × Y and S ⊆ Y × Z, x ∈ RS (A) if and only if S ◦ R[x] ⊆ A. NOTE ON PRIESTLEY-STYLE DUALITY FOR DISTRIBUTIVE MEET-SEMILATTICES 3

From the definition it follows that R is a homomorphism between semilattices Y∗ and X∗. In [2] it is shown that the category of generalized Priestley spaces and generalized Priestley morphisms, denoted by GPS, is dually equivalent to BDS. Remark 2.3. The identity morphism in GPS for a generalized Priestley space X is ≤X while the composition of generalized Priestley morphisms in GPS is given as follows: for X,Y and Z generalized Priestley spaces and R ⊆ X ×Y and S ⊆ Y ×Z generalized Priestley morphisms, the composition (S?R) ⊆ X × Z is given by: ∗ 3 (x, z) ∈ S?R iff (∀U ∈ Z ) if x ∈ RS(U), then z ∈ U. Let us conclude the review of the duality in [2] by showing how to get a general- ized Priestley space from an arbitrary bounded distributive meet-semilattice M and similarly for morphisms. We define the representation map ϕ : M −→ P(Op(M)) that assigns to each a ∈ M the collection ϕ(a) = {F ∈ Op(M): a ∈ F }. Then we take {ϕ(a): a ∈ M} ∪ {Op(M) \ ϕ(b): b ∈ M} as a subbasis for a topology τM on Op(M). Thus we get that hOp(M), τM, ⊆, Pr(M)i is a generalized Priestley space, called the Priestley dual of M. Let M1 and M2 be two bounded distributive meet-semilattices and let h : M1 −→ M2 be a homomorphism between them. We define the relation Rh ⊆ Op(M2) × Op(M1) as follows: (P,Q) ∈ Rh if and only if −1 h [P ] ⊆ Q. Thus we get that Rh is a generalized Priestley morphism, called the Priestley dual morphism of h. We move now to review the duality for bounded distributive meet-semilattices that can be obtain from the results in [3]. Recall that a is a to- pological space hX, τi that is compact, coherent and sober, where coherent means that KO(X), the collection of open compact subsets of X, is a basis that is closed under finite intersections, and sober means that every irreducible closed subset is the closure of a unique point. In [3] Celani studied a duality for distributive meet-semilattices with top element (but not necessarily with bottom), that can be easily restricted to a duality for bounded distributive meet-semilattices, as it is shown in Section 10 in [2]. The resulting duality for BDS is called in [2] a Spectral-like duality. However dual objects in this case are not Spectral spaces with additional structure. More precisely, they are compact, sober and compactly based, but they are not necessarily coherent. Following a similar strategy as in the Priestley case, a rightfully Spectral-like duality for meet-semilattices could be built from the Spectral dual of L(M), that can be also described in terms of optimal filters of M. It would be very interesting to go in depth into this duality, as it would fall midway between the duality in [2] and the duality in [3]. Regarding objects, Celani’s duality [3] is based on the representation theorem of distributive semilattices that goes back to Stone [6] and Gr¨atzer[5]. We supply here a simplified version of the original definition of dual spaces (Definition 14 in [3]), that was presented by Bezhanishvili and Jansana in [2]. Definition 2.4. (Definition in Section 10 in [2]) A pair X = hX, τi is a DS-space when it is a compactly-based sober topological space, this is, when:

3Notice that in [2] there is a notational inconsistency that is worth being aware of: up to p. 107, where ? is defined, the convention used for composition of relations is the one usually adopted in category theory, namely the left composition, where the first applied relation is the left one. From that point on, the convention used for both ◦ and ? is the usual one for composition of relations, namely the right composition. 4 MAR´IA ESTEBAN

(DS1) the collection KO(X) of open compact subsets forms a basis for the topo- logy τ, (DS2) X is sober. An alternative condition to (DS2) is: 0 (DS2 ) X is T0, compact and if Z is a closed subset and L is a non-empty down- directed subfamily of KO(X) such that Z ∩ U 6= ∅ for all U ∈ L, then Z ∩ T{U : U ∈ L}= 6 ∅. A CDS-space is a compact DS-space. Let us denote by X+ the collection of all complements of open compact subsets of X, i. e. {X \ U : U ∈ KO(X)}. From the definition it follows that X+ := hX+, ∩, ∅,Xi is a bounded distributive meet- semilattice, called the dual semilattice of X. The original definition of dual mor- phisms (Definition 22 in [3]) was also simplified by Bezhanishvili and Jansana in [2], and this is the one that we adopt. Definition 2.5. (Definition in Section 10 in [2]) Let X and Y be DS-spaces. A relation R ⊆ X × Y is called a meet-relation when: (DSR1) If U ∈ KO(Y ), then X \ (R(Y \ U)) ∈ KO(X), (DSR2) R[x] is a closed subset for each x ∈ X.

From the definition it follows that R is a homomorphism between semilattices Y+ and X+. From restuls in [3] it follows that the category of CDS-spaces and meet-relations, denoted by CDS, is dually equivalent to BDS. Remark 2.6. The identity morphism in CDS for a CDS-space X is the dual of the specialization order of X, denoted by ≤X , while the composition of morphisms in CDS as follows: for X,Y and Z CDS-spaces and R ⊆ X × Y and S ⊆ Y × Z meet-relations, the composition (S ∗ R) ⊆ X × Z is given by:  (x, z) ∈ S ∗ R iff ∀U ∈ KO(Z) x ∈ RS(X \ U) ⇒ z ∈ X \ U . Let us conclude this section by showing how to get a CDS-space from an arbitrary bounded distributive meet-semilattice M and similarly for morphisms. We define the representation map ψ −→ P(Pr(M)) that assigns to each a ∈ M the collection ψ(a) = {F ∈ Pr(M): a ∈ F }. Then we take {Pr(M) \ ψ(a): a ∈ M} as a subbasis for a topology τM on Pr(M). Thus we get that hPr(M), τMi is a CDS- space, called the Spectral dual of M. Let M1 and M2 be two bounded distributive meet-semilattices and let h : M1 −→ M2 be a homomorphism between them. We define the relation Rh ⊆ Pr(M2) × Pr(M1) as follows: (P,Q) ∈ Rh if and only −1 if h [P ] ⊆ Q. Thus we get that Rh is a meet-relation, called the Spectral dual morphism of h.

3. From the Priestley-style duality to the Spectral-like duality Bezhanishvili and Jansana showed in [2], how to obtain a CDS-space from a generalized Priestley space, and how to obtain a meet-relation from a generalized Priestley morphism. Let us just briefly review the construction they work with. On the one hand, for any generalized Priestley space X = hX, τ, ≤,X0i, the structure X0 := hX0, τ0i is a CDS-space, where τ0 is the topology on X0 with basis ∗ {X0 \ U : U ∈ X }. On the other hand, for any generalized Priestley spaces X and Y , and any generalized Priestley morphism R ⊆ X × Y , the relation R0 := R ∩ (X0 × Y0) is a meet-relation between CDS-spaces X0 and Y0. An important NOTE ON PRIESTLEY-STYLE DUALITY FOR DISTRIBUTIVE MEET-SEMILATTICES 5 fact, proved in [1], is that compact-opens of X0 are in one-to-one correspondence with X0-admissible clopen up-sets of X, by means of the following maps: ∗ ∗ α : KO(X0) −→ X ξ : X −→ KO(X0)

V 7−→ X \ (↓V ) U 7−→ X0 \ U In summary, from results in [2] and [1] we can define the F : GPS −→ CDS such that for any generalized Priestley spaces X,Y and any generalized Priestley morphism R ⊆ X × Y :

F(X) := hX0, τ0i

F(R) := R ∩ (X0 × Y0)

Notice that F(≤X ) =≤X0 =≤F(X), so the functor preserves the identity morphism for GPS. Let us check that it also preserves composition of mophisms in GPS. Lemma 3.1. Let X,Y,Z be generalized Priestley spaces and let R ⊆ X × Y and S ⊆ Y × Z be generalized Priestley morphisms. Then for any x ∈ X0 and any z ∈ Z0:

(x, z) ∈ (S?R) ∩ (X0 × Z0) iff (x, z) ∈ S0 ∗ R0.

∗ Proof. Let first (x, z) ∈ (S?R) ∩ (X0 × Z0), so for all U ∈ X , if S ◦ R[x] ⊆ U, then z ∈ U. Suppose, towards a contradiction, that (x, z) ∈/ S0 ∗ R0. Then there is V ∈ KO(Z0) such that S0 ◦ R0[x] ⊆ X0 \ V and z∈ / X0 \ V . Hence z ∈ V and so z∈ / X \ (↓V ), which is an X0-admissible clopen up-set of X. Therefore by assumption S ◦ R[x] * X \ (↓V ), and then there is z0 ∈ Z such that (x, z0) ∈ S ◦ R 0 00 0 00 00 and z ∈ ↓V . So there is z ∈ V ⊆ Z0 such that z ≤ z . But then z ∈ S0 ◦ R0[x], since ≤Z ◦ S = S, and then by assumption S0 ◦ R0[x] * X0 \ V , a contradiction. For the converse, let (x, z) ∈ S0 ∗ R0, so for all V ∈ KO(Z0), if S0 ◦ R0[x] ⊆ X0 \ V , then z ∈ X0 \V . Suppose, towards a contradiction, that (x, z) ∈/ (S?R)∩(X0 ×Z0). ∗ Then there is U ∈ X such that S ◦ R[x] ⊆ U and z∈ / U. Therefore S0 ◦ R0[x] ⊆ U ∩ X0 = X0 \ (X0 \ U), thus by assumption z ∈ U ∩ X0, a contradiction. 

4. From the Spectral-like duality to the Priestley-style duality As we have already shown, the definition of the functor from GPS to CDS is relatively simple to obtain. However, the definition of the functor that goes the other way around is considerably more involved. This construction was not studied in the cited works by Bezhanishvili and Jansana, and it is the purpose of the present section to study it. Let hX, τi be a CDS-space. First we show how to obtain a generalized Priest- ley space from X. Recall that the dual semilattice of X is X+ = hX+, ∩, ∅,Xi where X+ = {X \ U : U ∈ KO(X)}. Moreover, the Priestley dual of X+ is + + + + hOp(X ), τX+ , ⊆ Pr(X )i, where recall that Op(X ) (resp. Pr(X )) denotes the collection of optimal (resp. prime) filters of X+. We aim to provide a more direct construction of this structure, that is in fact equivalent to X. By convenience, for any U ⊆ X we will use U c as a shorthand for X \ U. Let us consider the collection of all closed subsets of X, C(X). Recall that order filters of X+ are in one-to-one correspondence with closed subsets of X by means of the following maps: 6 MAR´IA ESTEBAN

f : C(X) −→ Fi(X+) C 7−→ {U c : U ∈ KO(X),C ⊆ U c}

g : Fi(X+) −→ C(X) \ F 7−→ {U c : U ∈ KO(X),U c ∈ F } Proposition 4.1. For any CDS-space X = hX, τi, there is an order dual isomor- phism between hC(X), ⊆i and hFi(X+), ⊆i given by the maps f and g. + + Let us thorougly analyze the image by g of Pr(X ) and Op∧(X ) respectively. In what follows we will repeatedly use that for any C1,C2 ∈ C(X), f(C1)∩f(C2) = + f(C1 ∪ C2), and for any F1,F2 ∈ Fi(X ), g(F1) ∪ g(F2) = g(F1 ∩ F2). By sobriety we know that for each x ∈ X, the topological closure of x (the smallest closed set containing x is an irreducible closed subset of X and all irreducible closed subsets have this form. We denote by CIrr(X) the collection of all irreducible closed subsets. Recall that an irreducible closed subset is a closed subset B ∈ C(X) such that whenever B ⊆ C1 ∪ C2 for closed subsets C1 and C2, then either B ⊆ C1 or Irr B ⊆ C2. The map cl : X −→ C (X), that assigns to each element its topological closure, stablishes a dual order between X ordered by the dual of the specialization order, and hCIrr(X), ⊆i. Proposition 4.2. For any CDS-space X = hX, τi, there is an order dual isomor- phism between hCIrr(X), ⊆i and hPr(X+), ⊆i given by the maps f and g. Proof. First we show that for C an irreducible closed subset of X, f(C) is a prime + + filter of X . Let F1,F2 ∈ Fi(X ) be such that F1,F2 * f(C). Then C * g(F1) and C * g(F2), and by C being an irreducible closed subset, we get C * g(F1)∪g(F2) = + g(F1 ∩ F2), and therefore F1 ∩ F2 * f(C). Hence f(C) is a prime filter of X . Now we show that for F a prime filter of X+, g(F ) is an irreducible closed subset of X. Let C1,C2 ∈ C(X) be such that g(F ) * C1,C2. Then f(C1), f(C2) * F , and by F being prime f(C1 ∪ C2) = f(C1) ∩ f(C2) * F , and therefore g(F ) * C1 ∪ C2. Hence g(F ) is an irreducible closed subset of X.  Let us introduce now a concept that aims to captures the closed subsets that are in the image of Op(X+) by g. Definition 4.3. A closed subset C ∈ C(X) is optimal when for all V ∈ KO(X) and all finite U ⊆ KO(X), if T U ⊆ V and C ∩U 6= ∅ for all U ∈ U, then C ∩V 6= ∅. We denote by COp(X) the collection of all optimal closed subsets of X. Notice that since X ∈ KO(X), from the definition it follows that optimal closed subsets are non-empty. Hence, an alternative definition is: C ∈ C(X) is optimal when it is non-empty and for all n ∈ ω and V,U0,...,Un ∈ KO(X), if U0 ∩ · · · ∩ Un ⊆ V and Irr Op C ∩ Ui 6= ∅ for all i ≤ n, then C ∩ V 6= ∅. Clearly we have C (X) ⊆ C (X). Proposition 4.4. For any CDS-space X = hX, τi, there is an order dual isomor- phism between hCOp(X), ⊆i and hOp(X+), ⊆i given by the maps f and g. Proof. First we show that for C an optimal closed subset of X, f(C) is an optimal filter of X+, by showing that X+ \ f(C) is a Frink ideal of X+. Since C is non- empty, let x ∈ C. Then by KO(X) being a basis for X, there is U ∈ KO(X) such NOTE ON PRIESTLEY-STYLE DUALITY FOR DISTRIBUTIVE MEET-SEMILATTICES 7

c + that x ∈ U, and so C * U . Therefore X \ f(C) is non-empty. Let U1,...,Un ∈ c c c c c KO(X), for some n ∈ ω be such that U1 ,...,Un ∈/ f(C) and ↑U1 ∩· · ·∩↑Un ⊆ ↑W for some W ∈ KO(X). Suppose, towards a contradiction, that W c ∈ f(C), i. e. C ∩ W = ∅. As by assumption C ∩ Ui 6= ∅ for all i ≤ n and since C is an optimal closed subset we get U1 ∩ · · · ∩ Un * W . So there is x ∈ (U1 ∩ · · · ∩ Un) \ W . By KO(X) being a basis, there is V ∈ KO(X) such that x ∈ V ⊆ U1 ∩ · · · ∩ Un. Thus c c c c c V * W , and so we have V ∈ ↑U1 ∩ · · · ∩ ↑Un and V ∈/ ↑W , a contradiction. Now we show that for P an optimal filter of X+, g(P ) is an optimal closed subset of X. By definition P is proper, so ∅ ∈/ P . Suppose, towards a contradiction, that c c g(P ) = ∅. Then by compactness of the space, there are U1 ,...,Un ∈ P , for some c c + n ∈ ω, such that U1 ∩ · · · ∩ Un = ∅. But since P is a filter of X , then ∅ ∈ P , a contradiction. Hence g(P ) is non-empty. Let now U1,...,Un,V ∈ KO(X), for some n ∈ ω, be such that g(P ) ∩ Ui 6= ∅ for all i ≤ n and U1 ∩ · · · ∩ Un ⊆ V . Notice that for all U ∈ KO(X), U c ∈ P if and only if g(P ) ∩ U = ∅. Therefore, c by assumption we have Ui ∈/ P for all i ≤ n. Suppose, towards a contradiction, that g(P ) ∩ V = ∅. Then V c ∈ P , and by P being an optimal filter, we conclude c c c c c ↑U1 ∩ · · · ∩ ↑Un * ↑V . So let W ∈ KO(X) be such that Ui ⊆ W for all i ≤ n c c and V * W . Then we get W ⊆ U0 ∩ · · · ∩ Un ⊆ V , a contradiction.  Lemma 4.5. For any CDS-space X = hX, τi and for any non-empty U ⊆ KO(X), the Frink ideal of X+ generated by {U c : U ∈ U} is c c c c {W : W ∈ KO(X), (∃n ∈ ω)(U1,...,Un ∈ U) W ⊆ U1 ∪ · · · ∪ Un}. c Proof. Let U ⊆ KO(X) and let Z := {W : W ∈ KO(X), (∃n ∈ ω)(U1,...,Un ∈ c c c U) W ⊆ U1 ∪ · · · ∪ Un}. Clearly Z is included in the Frink ideal generated by {U c : U ∈ U}. For the reverse inclusion, let V c be an element in that Frink ideal, c c c so there are n ∈ ω and U1,...,Un ∈ U such that ↑U1 ∩ · · · ∩ ↑Un ⊆ ↑V . Suppose, c c c c c c towards a contradiction, that V * U1 ∪ · · · ∪ Un, and let x ∈ V \ (U1 ∪ · · · ∪ Un). c c Since U1 ∪· · ·∪Un is a closed subset, by KO(X) being a basis, there is W ∈ KO(X) c c c c c c c such that U1 ∪ · · · ∪ Un ⊆ W and x∈ / W . Hence W ∈ ↑U1 ∩ · · · ∩ ↑Un and c c W ∈/ ↑V , a contradiction.  Now we have all we need to define the generalized Priestley space we are looking for. We need to add more points to the ones that we have in our CDS-space. For Op Irr each C ∈ C (X) \C (X), we add a new point xC to the collection X. Then we obtain the collection Op Irr X := X ∪ {xC : C ∈ C (X) \C (X)}. Irr For convenience, sometimes we refer to any x ∈ X by xcl(x). As cl(x) ∈ C (X) ⊆ Op C (X), we further assume that all elements in X have the form xC for some C ∈ Op C (X). Moreover, an order can be defined on X as follows: for each xC , xC0 ∈ X 0 xC ≤ xC0 iff C ⊆ C. Notice that this order extends the dual of the specialization order of X. Let us consider the map η : KO(X) −→ P(X) given by: c η(U) := {xC ∈ X : C ⊆ U }. For each U ∈ KO(X), we denote by η(U)c the set X \ η(U). Consider the topology τ on X having as subbasis the collection {η(U): U ∈ KO(X)} ∪ {η(V )c : V ∈ KO(X)}. 8 MAR´IA ESTEBAN

Remark 4.6. Notice that from KO(X) being closed under finite unions, it follows that the collection {η(U): U ∈ KO(X)} is closed under finite intersections, and therefore, for any U1,...,Un ∈ KO(X), η(U1) ∩ · · · ∩ η(Un) = η(U1 ∪ · · · ∪ Un). We claim that the structure hX, τ, ≤,Xi is a generalized Priestley space. We prove the claim by showing that there is an order homeomorphism f between + + hX, τ, ≤i, and hOp(X ), τX+ , ⊆i, such that f[X] = Pr(X ). We define the map h : X −→ Op(X+) such that: c + c h(xC ) := f(C) = {U ∈ X : C ⊆ U }. By Proposition 4.4 and definition of X we know that the map h is well-defined, and + −1 that it is in fact an isomorphism, such that for each P ∈ Op(X ), h (P ) = xg(P ). Moreover, from the definition of the order in X, we get that h is order preserving. + Recall that the topology τX+ on Op(X ) is defined (see p. 99 in [2]) from the subbasis {ϕ(U c): U ∈ KO(X)} ∪ {ϕ(U c)c : U ∈ KO(X)}, where ϕ : X+ −→ P(Op(X+)) is given by: ϕ(U c) = {P ∈ Op(X+): U c ∈ P }. Notice that for any P ∈ Op(X+): c c c P ∈ ϕ(U ) iff U ∈ P iff g(P ) ⊆ U iff xg(P ) ∈ η(U) iff P ∈ h[η(U)]. Therefore, h[η(U)] = ϕ(U c) for all U ∈ KO(X). From this fact it follows that h + is a continuous and open function between hX, τi and hOp(X ), τX+ i, and hence it is a homeomorphism. It follows also that h[η[KO(X)]] = ϕ[X+]. Furthermore, for any x ∈ X, h(x) = f(cl(x)), and then by Proposition 4.2 we conclude that h[X] = Pr(X+). Corollary 4.7. For any CDS-space X = hX, τi, the map h is an order homeomor- + + phism between hX, τ, ≤i, and hOp(X ), τX+ , ⊆i, such that h[X] = Pr(X ). Corollary 4.8. For any CDS-space X = hX, τi, the structure hX, τ, ≤,Xi is a generalized Priestley space. The previous corollary gives us how the functor we are looking for acts on the objects of CDS. Now we move to morphisms. Let hX1, τ1i and hX2, τ2i be CDS- spaces and let R ⊆ X1 × X2 be a meet-relation between them. We define the relation R¯ ⊆ X1 × X2 as follows: ¯ c c (xC1 , xC2 ) ∈ R iff (∀V ∈ KO(X2)) if R[C1] ⊆ V , then C2 ⊆ V , S where recall that R[C1] = {R[x]: x ∈ C1}. We claim that R¯ is a general- ized Priestley morphism between generalized Priestley spaces hX1, τ 1, ≤,X1i and hX2, τ 2, ≤,X2i. In order to show this, we prove first some useful lemmas, that moreover give us the characterization of open up-sets, clopen up-sets and admissi- ble clopen up-sets of the generalized Priestley space X associated with an arbitrary CDS-space X. Lemma 4.9. Let X = hX, τi be a CDS-space. Each open up-set of the generalized Priestley space hX, τ, ≤,Xi is a union of elements of {η(U): U ∈ KO(X)}. NOTE ON PRIESTLEY-STYLE DUALITY FOR DISTRIBUTIVE MEET-SEMILATTICES 9

Proof. Let W be an open up-set of hX, τ, ≤,Xi. We show that for each xC ∈ W there is U ∈ KO(X) such that xC ∈ η(U) ⊆ W . Let xC ∈ W . Then for each 0 xC0 ∈/ W , xC  xC0 and so C * C. By Priestley separation axiom, there is T UC0 ∈ KO(X) such that xC ∈ η(UC0 ) and xC0 ∈/ η(UC0 ). Hence {η(UC0 ): xC0 ∈/ c W } ∩ W = ∅, and by compactness, we get U0,...,Un ∈ KO(X), for some n ∈ ω, such that xC ∈ η(U0 ∪ · · · ∪ Un) = η(U0) ∩ · · · ∩ η(Un) ⊆ W , as required.  Lemma 4.10. Let X = hX, τi be a CDS-space. Each clopen up-set of the gen- eralized Priestley space hX, τ, ≤,Xi is a finite union of elements of {η(U): U ∈ KO(X)}.

Proof. This follows from previous lemma and compactness of the space.  Lemma 4.11. Let X = hX, τi be a CDS-space. A subset W ⊆ X is an X- admissible a clopen up-set of the generalized Priestley space hX, τ, ≤,Xi if and only if W = η(U) for some U ∈ KO(X). Proof. Let first U ∈ KO(X). By definition, η(U) is clopen a up-set of hX, τ, ≤,Xi. c c 0 c Let xC ∈ max(η(U) ). So we have that C * U and for all xC0 ∈ X, if C * U , c then xC ≮ xC0 . As there is x ∈ C \ U , on the one hand, cl(x) ⊆ C, i. e. xC ≤ x, c and on the other hand cl(x) * U , which implies by assumption xC ≮ x. Hence c x = xC . We conclude that max(η(U) ) ⊆ X, and so η(U) is an X-admissible clopen up-set of X, as required. Let now W be an X-admissible clopen up-set of hX, τ, ≤,Xi. By Lemma 4.10, there are U1,...,Un ∈ KO(X) such that W = η(U1) ∪ · · · ∪ η(Un). Consider the c c closed subset D := U1 ∪· · ·∪Un, and the filter associated with it f(D). Consider also c c c c the subset J := {W : W ∈ KO(X),W ⊆ U1 ∪ · · · ∪ Un}, that is, by Lemma 4.5, c c the Frink ideal generated by {U1 ,...,Un}. We claim that f(D) ∩ J 6= ∅. Suppose not, then by the optimal filter lemma (Lemma 4.7 in [2]), there is an optimal filter c P such that f(D) ⊆ P and J ∩ P = ∅. Then for each i ≤ n, since Ui ∈ J, we have c c Ui ∈/ P , i. e. g(P ) * Ui . Therefore xg(P ) ∈/ η(U1) ∪ · · · ∪ η(Un) = W . By W being X-admissible, there is x ∈ X such that xg(P ) ≤ x∈ / W . Therefore we have cl(x) ⊆ c c g(P ). Moreover, since f(D) ⊆ P , then g(P ) ⊆ D, and so cl(x) ⊆ D = U1 ∪· · ·∪Un. c c Thus x ∈ Ui for some i ≤ n, and so cl(x) ⊆ Ui , i. e. x = xcl(x) ∈ η(Ui) ⊆ W , a contradiction. We conclude that f(D) ∩ J 6= ∅. Then there is V ∈ KO(X) such c c c that U1 ∪· · ·∪Un = V , and therefore, using the definition of optimal closed subset, we get W = η(V ), as required.  By the previous lemma we conclude that the map η gives us a one-to-one corres- pondence between KO(X) and (X)∗, the collection of X-admissible clopen up-sets of X.

Proposition 4.12. For any meet-relation R ⊆ X1 × X2 between CDS-spaces hX1, τ1i and hX2, τ2i and for any x1 ∈ X1 and x2 ∈ X2:

(x1, x2) ∈ R iff (x1, x2) ∈ R.¯ ¯ Moreover for all xC1 ∈ X1 and all x2 ∈ R[C1], it holds that (xC1 , x2) ∈ R.

Proof. Recall that for each x ∈ X, we identify xcl(x) with x, so for the first statement, the inclusion from left to right is immediate. For the converse, as- sume (x1, x2) ∈ R¯ and suppose, towards a contradiction, that (x1, x2) ∈/ R. So x2 ∈/ R[x1], that is a closed subset of X2 by condition (DSR2). Then there 10 MAR´IA ESTEBAN

c c is U ∈ KO(X2) such that x2 ∈ U and R[x1] ⊆ U . From x2 ∈/ U , we get c c c cl(x2) * U , and then by hypothesis, R[cl(x1)] * U . From R[x1] ⊆ U , we get c c x1 ∈ R(U ), that is a closed subset by (DSR1), and so cl(x1) ⊆ R(U ). It follows c that R[cl(x1)] ⊆ U , a contradiction. The second statement follows easily. 

Proposition 4.13. For any meet-relation R ⊆ X1 × X2 between CDS-spaces hX1, τ1i and hX2, τ2i and for any xC1 ∈ X1: ¯ c R[xC1 ] ⊆ η2(U) iff R[C1] ⊆ U . ¯ Proof. Assume first that R[xC1 ] ⊆ η2(U) and let x2 ∈ R[C1]. Then by Proposition ¯ 4.12 we have (xC1 , x2) ∈ R, and then by assumption x2 ∈ η2(U). Thus by definition c c c of η2, cl(x2) ⊆ U , and hence x2 ∈ U . For the converse, assume R[C1] ⊆ U and ¯ ¯ c let xC2 ∈ R[xC1 ]. Then by the definition of R, C2 ⊆ U , i. e. xC2 ∈ η2(U), as required. 

Theorem 4.14. For any meet-relation R ⊆ X1 × X2 between CDS-spaces hX1, τ1i and hX2, τ2i, the relation R¯ is a generalized Priestley morphism between generalized Priestley spaces hX1, τ 1 ≤,X1i and hX2, τ 2 ≤,X2i.

Proof. First we show that condition (PrR2) holds, i. e. we have to show that R¯(W ) ∗ is an X1-admissible clopen upset of X1 for any W ∈ (X2) . By Lemma 4.11, it c c is enough to show that R¯(η2(U)) = η1(R(U ) ) for all U ∈ KO(X2). And by Proposition 4.13 we get that for any U ∈ KO(X2): ¯ c xC1 ∈ R¯(η2(U)) iff R[xC1 ] ⊆ η2(U) iff R[C1] ⊆ U c c c iff C1 ⊆ R(U ) iff xC1 ∈ η1((R(U )) ). Now we show that condition (PrR1) also holds, i. e. we have to show that if ¯ ∗ ¯ (xC1 , xC2 ) ∈/ R, then there is W ∈ (X2) such that xC2 ∈/ W and R[xC1 ] ⊆ W . ¯ By Lemma 4.11 again, it is enough to show that if (xC1 , xC2 ) ∈/ R, then there ¯ is U ∈ KO(X2) such that xC2 ∈/ η2(U) and R[xC1 ] ⊆ η2(U). Let xC1 ∈ X1 ¯ ¯ and xC2 ∈ X2 and assume (xC1 , xC2 ) ∈/ R. Then by definition of R, there is c c U ∈ KO(X2) such that R[C1] ⊆ U and C2 * U . This implies xC2 ∈/ η2(U) and ¯ R[xC1 ] ⊆ η2(U), as required. 

In summary, from above results we can define the functor G : CDS −→ GPS such that for any CDS-spaces X,Y and any meet-relation R ⊆ X × Y :

GX) := hX, τ, ≤,Xi G(R) := R¯

It is only left to show that G is well-defined. For any CDS-space X, ≤X is the identity morphism for X while ≤X is the identity morphism for X. It follows from (Pr5) that G(≤X ) =≤X . Let us check that G preserves composition of morphisms. Lemma 4.15. Let X,Y,Z be CDS-space and let R ⊆ X × Y and S ⊆ Y × Z be meet-relations. Then for any xC ∈ X and any xC0 ∈ Z:

(xC , xC0 ) ∈ (S ∗ R) iff (xC , xC0 ) ∈ S? R.¯ NOTE ON PRIESTLEY-STYLE DUALITY FOR DISTRIBUTIVE MEET-SEMILATTICES 11

c Proof. Let xC ∈ X. First we show that for any U ∈ KO(Z), we have S ◦R[C] ⊆ U if and only if S ◦ R¯[xC ] ⊆ ηZ (U). Notice that by definition: c c c S ◦ R[C] * U iff C * RS(U ) iff R[C] * S(U ) ¯ 0 c iff (∃xC0 ∈ R[xC ]) S[C ] * U ¯ 00 c iff (∃xC0 ∈ R[xC ])(∃xC00 ∈ S[xC0 ]) C * U ¯ iff S ◦ R[xC ] * ηZ (U). From this fact it follows that G preserves composition of morphisms, as we get that for any xC0 ∈ Z: c 0 c (xC , xC0 ) ∈/ (S ∗ R) iff (∃U ∈ KO(Z)) S ∗ R[C] ⊆ U and C * U c 0 c iff (∃U ∈ KO(Z)) S ◦ R[C] ⊆ U and C * U

iff (∃U ∈ KO(Z)) S ◦ R¯[xC ] ⊆ ηZ (U) and xC0 ∈/ ηZ (U)

iff (xC , xC0 ) ∈/ S? R.¯  5. Categorical duality We finally introduce the natural involved in the equivalence of the categories CDS and GPS. Let us consider first the endofunctor FG : CDS −→ CDS. Theorem 5.1. FG is the identity functor in CDS. Proof. This follows easily from the definitions of F and G. On the one hand, for any CDS-space X = hX, τi, we have that FG(X) = hX, (τ)0i, where (τ)0 has as basis {X \ η(U): U ∈ KO(X)}. Note that for each U ∈ KO(X) we have: c c X\η(U) = X\{xC ∈ X : C ⊆ U } = {x ∈ X : cl(x) * U } = {x ∈ X : x ∈ U} = U, ¯ ¯ hence (τ)0 = τ. On the other hand we have that (R)0 = R ∩ (X × X) = R. 

For any CDS-space X = hX, τi, let ≤X be the order associated with it, namely the dual of the specialization order of the space. We know that ≤X is the identity morphism for X in CDS, and so it is an isomorphism in CDS. Consider the family of morphisms in CDS:

Φ := (≤X ⊆ X × FG(X))X∈CDS Theorem 5.2. Φ is a natural isomorphism between the identity functor in CDS and FG.

Proof. Let X,Y be CDS-spaces and let R ⊆ X × Y be a meet-relation. By ≤X being the identity morphism in X and by Theorem 5.1, we get (R¯)0 ∗ ≤X = (R¯)0 = R = ≤Y ∗ R, so we are done.  The previous corollary gives us one of the required natural isomorphisms. Let us move to consider the endofunctor GF : GPS −→ GPS. We need to show that for each generalized Priestley space X = hX, τ, ≤,X0i, there is an isomorphism SX between X and GF(X) such that for each generalized Priestley morphism R ⊆ X × Y , SY ?R = GF(R) ?SX . Let X = hX, τ, ≤,X0i be a generalized Priestley space. For any X0-admissible c clopen up-set U, by η(X0 \ U) we denote the set {xC ∈ (X0): C * U ∩ X0}. 12 MAR´IA ESTEBAN

Lemma 5.3. Let X = hX, τ, ≤,X0i be a generalized Priestley space. For any Op Op x ∈ X, we have that ↑x ∩ X0 ∈ C (X0). Moreover, for each C ∈ C (X0), there is x ∈ X such that ↑x ∩ X0 = C.

Proof. First we check that ↑x ∩ X0 is a closed subset of X0. Let y ∈ X0 be such ∗ that y∈ / ↑x ∩ X0. By assumption x 6= y, so by (Pr5) there is U ∈ X such that x ∈ U and y∈ / U. Hence y ∈ X0 \ U and (↑x ∩ X0) ∩ (X0 \ U) = ∅. As basic opens ∗ of X0 have the form X0 \ U for some U ∈ X , we are done. We show now that ↑x∩X0 is an optimal closed subset of X0. By (Pr3) we get that ∗ it is non-empty. Let n ∈ ω and V,U0,...,Un ∈ X , be such that (X0 \ U0) ∩ · · · ∩ (X0 \Un) ⊆ X0 \V , and for all i ≤ n,(↑x∩X0)∩(X0 \Ui) 6= ∅. This implies that for each i ≤ n, x∈ / Ui. We claim that (↑x∩X0)∩(X0\V ) 6= ∅. Suppose not, then we get x∈ / V , since otherwise, as V is X0-admissible, there is y ∈ X0 such that x ≤ y∈ / V , c c and then y ∈ (↑x ∩ X0) ∩ (X0 \ V ), a contradiction. Thus x ∈ V ∩ U0 ∩ · · · ∩ Un, c c that is an open subset. By density, there is z ∈ X0 ∩ (V ∩ U0 ∩ · · · ∩ Un). Then by assumption, from z∈ / X0 \ V , we get that there is i ≤ n such that z∈ / X0 \ Ui, i. e. z ∈ Ui, a contradiction. Op Consider now a closed subset C ∈ C (X0) and suppose, towards a contra- T ∗ T c ∗ diction, that {V ∈ X : C ⊆ V } ∩ {U : U ∈ X ,C * U} = ∅. Then since the elements of X∗ are clopen subsets of X, by compactness of X, there are ∗ V0,...,Vn,U0,...,Um ∈ X such that C ⊆ Vi, C * Uj, for all i ≤ n and j ≤ m, c c and V0 ∩ · · · ∩ Vn ∩ U0 ∩ · · · ∩ Um = ∅. Let V := V0 ∩ · · · ∩ Vn. Clearly C ⊆ V , and by assumption V ⊆ U0 ∪ · · · ∪ Um. Thus (X0 \ U0) ∩ · · · ∩ (X0 \ Um) ⊆ X0 \ V , and by C being an optimal closed subset, we conclude C ∩ (X0 \ V ) 6= ∅, a contra- T ∗ T c ∗ diction. Therefore, there is x ∈ {V ∈ X : C ⊆ V } ∩ {U : U ∈ X ,C * U}. 0 T ∗ Clearly ↑x ∩ X0 = C, as required. Moreover, if there are x, x ∈ {V ∈ X : C ⊆ T c ∗ 0 V } ∩ {U : U ∈ X ,C * U}, then by (Pr5) we get x = x , therefore there is a unique x ∈ X such that ↑x ∩ X0 = C.  The previous lemma gives us the following bijection between the elements of X and the elements of (X0): each y ∈ X corresponds with x↑y∩X0 ∈ (X0), and all elements of (X0) have this form. The following lemma is then easy to prove.

Lemma 5.4. Let X = hX, τ, ≤,X0i be a generalized Priestley space. For any ∗ U ∈ X , η(X0 \ U) = U.

Proof. Recall that by definition η(X0 \ U) = {xC ∈ (X0): C ⊆ U ∩ X0} = {x ∈ ∗ X : ↑x ∩ X0 ⊆ U ∩ X0} for any U ∈ X . Let first x ∈ η(X0 \ U), and suppose, c towards a contradiction, that x∈ / U. Then x ∈ U , and by U being X0-admissible, let y ∈ X0 such that x ≤ y, i. e. y ∈ ↑x ∩ X0. By assumption we obtain y ∈ U ∩ X0, a contradiction. We have proved that η(X0 \ U) ⊆ U. The converse follows easily, since for x ∈ U, as U is an up-set, we have ↑x ⊆ U, and so ↑x ∩ X0 ⊆ U ∩ X0, as required.  According to the previous lemma, the map η gives us a one-to-one correspondence between the subbasis of X, {U : U ∈ X∗} ∪ {V c : V ∈ X∗}, and the subbasis of ∗ c ∗ (X0), {η(X0 \ U): U ∈ X } ∪ {η(X0 \ V ) : V ∈ X }. Thus, the spaces hX, τi and h(X0), (τ0)i are homeomorphic. Similarly, for any generalized Priestley morphism ¯ R ⊆ Y × Z, it follows that (y, z) ∈ R if and only if (x↑y∩Y0 , x↑z∩Z0 ) ∈ R. For each generalized Priestley space X, let SX ⊆ X × (X0) be the relation given by:

(x, x↑y∩X0 ) ∈ SX iff x ≤ y. NOTE ON PRIESTLEY-STYLE DUALITY FOR DISTRIBUTIVE MEET-SEMILATTICES 13

It is easy to check that SX is a generalized Priestley morphism, and it is in fact an isomorphism in GPS. Consider the family of morphisms in GPS:

Ψ = (SX ⊆ X × GF(X))X∈GPS Lemma 5.5. Let X,Y be generalized Priestley spaces and let R ⊆ X × Y be a generalized Priestley morphism. Then (x, y) ∈ R if and only if (x, y) ∈ (R0). Proof. Assume first that (x, y) ∈/ R. Then by (PrR1), there is U ∈ Y ∗ such that y∈ / U and R[x] ⊆ U. Thus ↑y ∩ Y0 * U ∩ Y0 and R0[↑x ∩ X0] ⊆ U ∩ Y0. Therefore, using that ↑x∩X0 and ↑y ∩Y0 are optimal closed subsets of X0 and Y0 respectively, and that they correspond with the elements x ∈ (X0) and y ∈ (Y0) respectively, from the definition of (R0) we get (x, y) ∈/ (R0). Assume now that (x, y) ∈ R and suppose, towards a contradiction, that (x, y) ∈/ ∗ (R0). Then by (PrR1), there is U ∈ Y such that y∈ / η(Y0 \ U) and (R0)[x] ⊆ η(Y0 \ U). By definition of η we obtain that ↑y ∩ Y0 * U ∩ Y0, so y∈ / U. By assumption y ∈ R[x], so R[x] * U, i. e. x∈ / R(U), which is an X0-admissible 0 0 0 clopen up-set of X. So there is x ∈ ↑x ∩ X0 such that x ∈/ R(U), i. e. R[x ] * U. 0 0 Then there is y ∈ R[x ] \ U, and since U is Y0-admissible, we can assume, without 0 0 0 loss of generality, that y ∈ Y0. Therefore we have y ∈/ U ∩Y0, and so y ∈/ η(Y0 \U). 0 0 0 0 Moreover, since x ∈ ↑x ∩ X0 and y ∈ R[x ], we get y ∈ R0[↑x ∩ X0], which implies 0 0 y ∈ (R0)[x]. Hence y witnesses that (R0)[x] * η(Y0 \ U), a contradiction.  By this lemma we conclude that for any generalized Priestley morphism R, (R0) = R. Moreover, using (Pr5) and the definition of ≤ , it follows that for (X0) any x, y ∈ X, x ≤ y iff ↑y ∩ X0 ⊆ ↑x ∩ X0 iff x ≤ y. (X0) Remark 5.6. To be precise, GF is not the identity functor in GPS, although it seems to be so, due to our notational convention according which we take (X0) = X. Theorem 5.7. Ψ is a natural isomorphism between the identity functor in GPS and GF. Proof. Let X,Y be generalized Priestley spaces and R ⊆ X × Y a generalized Priestley morphism. By ≤X being the identity morphism in X and by Lemma 5.5, we get (R0) ?SX = (R0) ? ≤X = (R0) = R = ≤Y ?R = SY ?R, so we are done.  Corollary 5.8. The categories GPS and BDS are equivalent by means of the func- tors F and G and the natural equivalences Ψ and Φ. Moreover FG is the identity functor in BDS.

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Departament de Logica,` Historia` i Filosofia de la Ciencia,` Facultat de Filosofia, Universitat de Barcelona (UB). Montalegre 6, 08001 Barcelona, Spain