SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM SERGIO A. CELANI AND MARÍA ESTEBAN Abstract. Distributive Hilbert Algebras with infimum, or DH -algebras, are algebras with implication and conjunction that are not the residuum of each other. These algebras do not fall under the scope of the usual duality theory for lattices expansions, precisely because of the lack of residuation. We propose a new approach, that consists on regarding the meet as the additional operation, and representing it by means of a subset of the dual space. In this paper we introduce a special kind of spaces, based on compactly based sober topological spaces. We prove that the category of these spaces, together with certain kind of relations, is dually equivalent to the category of DH -algebras and -semihomomorphisms. We restrict this result to give a duality for the category of DH -algebras and -homomorphisms. This duality generalizes the one presented in [4] for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for these algebras, namely (irreducible) meet filters, (irreducible) implicative filters and absorbent filters. 1. Introduction Stone s duality for Boolean algebras [20] has been generalised to distributive lattices in at least three ways (cf. [1] and its references). The approach initiated by Stone himself [21] leads to a representation in terms of compactly-based sober spaces in which the collection of compact open sets is closed under finite intersections. Further generalizations of this approach lead to dualities for distributive meetsemilattices [5, 15], implicative semilattices [3], Hilbert algebras [8] and Hilbert algebras with supremum [7]. What they all have in common is that they provide representations in terms of compactly-based sober spaces. We refer to this class of dualities as spectral-like. A different approach initiated by Priestley [19] leads to a representation in terms of ordered Hausdorff topological spaces. Although both approaches have been followed to generalize the pioneering work on representation of Boolean algebras with operators [17], the later has been held to be advantageous [6, 13], specially in view of recent developments of the theory of canonical extensions (see [14] and his references). The keypoint of this theory is that the additional n-ary operations either preserve joins (meets) in each coordinate, or send joins (meets) in each coordinate to meets (joins). Such n-ary operations are represented in terms of n + 1-ary relations satisfying certain conditions. In the present paper we focus our attention on a class of algebras that is not within the scope of these studies, since the additional operations do not preserve Date: March 31, 2013. 2010 Mathematics Subject Classification. Primary: 03G25; 06F35. Key words and phrases. Hilbert algebras, meet-semilattices, topological representation. 1

2 SERGIO A. CELANI AND MARÍA ESTEBAN nor reverse joins or meets in each coordinate. The largest class of algebras to which our study can be applied, is distributive meet-semilattices extended with a Hilbert-implication, such that the meet and the implication define the same order, but the later is not necesarily the right residual of the former. We call these algebras distributive Hilbert algebras with infimum (or DH -algebras). The lack of residuation is precisely what forces us to search a completly different route for a topological representation of this class of algebras, that will be suported in already existing dualities for distributive meet-semilattices and Hilbert algebras. In the present paper we provide a spectral-like duality for two categories based on DH - algebras. The strategy consists in looking at the meet operation as an additional operation on the Hilbert algebra, instead of what is customary, namely looking at the implication as an additional operation on the (semi)-lattice structure. Accordingly, the meet is represented by a subset satisfying certain conditions, instead of by a relation. A parallel study involving a Priestley-style duality for the same class of algebras will be developed in a forthcoming paper. The organization of the paper goes as follows. In Section 2 we present the preliminaries and we establish the basic notational conventions. Particularly, we recall the spectral-like duality for distributive meet-semilattices given in [5] and [15], the spectral-like duality for Hilbert algebras developed in [8] and [7], and we introduce the class of DH -algebras. In Section 3 we examine different notions of filters associated with a DH -algebra, and the relations between them, that yield the keypoint of our representation strategy. In Section 4 we present the duality for objects, where duals of DH -algebras are certain kind of spectral-like spaces augmented with a subset that satisfies some conditions. In Section 5 we extend the duality for morphisms between DH -algebras. Following [8], we deal with two different notions of morphism, namely, the usual algebraic notion of homomorphism and a weaker notion related to that of semi-homomorphism between Hilbert algebras. Then two categories are defined and the dual equivalences of categories is proved. Finally in Section 6 a topological characterization of the main classes of filters is given. 2. Preliminaries Let X, be a poset. For each Y X, Y denotes the up-set generated by Y, i.e. Y := {x X : y Y (y x)}. Similarly, Y denotes the down-set generated by Y. For x X, we use x and x as a shorthand for {x} and {x}. By P (X) we denote the collection of all up-sets of X,. For each Y X, Y c denotes the complement of Y, namely {x X : x / Y }. The symbol ω denotes the set of all natural numbers. Let L be a lattice. Recall that an element a L is meet irreducible when for all b, c L, if a = b c then a = b or a = c. Similarly, an element a is meet prime when for all b, c L, if b c a then b c or c a. It is known that when L is a distributive lattice, then prime an irreducible elements coincide. Let X 1, X 2 be two sets and let R X 1 X 2 be a binary relation relation between them. For any x 1 X 1, R(x 1 ) = {x 2 X 2 : (x 1, x 2 ) R}, and for each Y X 2, R 1 (Y ) = {x 1 X 1 : y Y ((x 1, y) R)}. For given sets X 1, X 2, X 3, functions f : X 1 X 2, g : X 2 X 3 and relations R X 1 X 2 and S X 2 X 3, the composition will be denoted by g f and S R respectively.

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM 3 Let X, τ be a topological space. When no confusion is possible, we will refer to it simply by X. For each Y X, cl(y ) denotes the closure of Y. Recall that a subset Y X is saturated when it is an intersection of open sets. The saturation of a subset Y is the smallest saturated set containing Y and it is denoted by sat(y ). We recall also that the specialization pre-order of X, τ is given by x X y iff x cl({y}) = cl(y). We denote by O(X) (C(X)) the collection of all open (closed) subsets of X and by KO(X) the collection of all open and compact subsets of X. We recall that Y X is irreducible if Y = Y 1 Y 2 implies Y = Y 1 or Y = Y 2 for any closed subsets Y 1, Y 2 C(X). The space X is sober if each closed irreducible subset is the closure of a unique point. Distributive meet-semilattices. Let us recall that a meet-semilattice with top element 1, is an algebra A = A,, 1 of type (2, 0) such that the operation is idempotent, commutative, associative, and a 1 = a, for all a A. As usual, the binary relation defined by a b iff a b = a is a partial order. In what follows we will say semilattice instead of meet-semilattice with top element 1. An order ideal of a semilattice A is a non-empty up-directed down-set of A, i.e. a down-set I A, I such that if a, b I, then there is c I such that a, b c. We denote by Id(A) the collection of all order ideals of A. Notice that all principal down-sets are, in particular, order ideals. A meet filter of a semilattice A is a non-empty up-set closed under the meet operation, i.e. an up-set F A such that 1 F and if a, b F, then a b F. Notice that all principal up-sets are, in particular, meet filters. A meet filter F is proper when F A. We denote by Fi(A) the collection of all meet filters of A. The set Fi(A) is closed under arbitrary intersections, so for any B A, we denote by B the least meet filter containing B. We call B the meet filter generated by B. It is well known that B = {a A : b 1,..., b n B (b 1 b n a)}. Notice that for all a A, a = a. We consider the bounded lattice Fi(A) := Fi(A),,, A,, where the meet operation is given by intersection and the join operation is given by the meet filter generated by the union. We say that a meet filter F of a semilattice A is -irreducible when it is a meet irreducible element of the lattice Fi(A). We denote by Irr (A) the collection of all -irreducible meet filters of A. Definition 2.1. A semilattice A is distributive when for every a, b, c A such that a b c, there are a, b A, with a a, b b and c = a b. A representation theorem for distributive semilattices can be obtained from Stone s work in [21], or more detailed in [15], where Grätzer considers distributive semilattices as a general framework to discuss topological representations of distributive lattices [15]. He studies some elementary properties distributive semilattices, one being that a semilattice A is distributive iff its lattice of filters Fi(A) is a distributive lattice. A categorical duality for distributive semilattices was studied in [5], where dual objects are topological spaces called DS-spaces. Recall that X = X, τ is a DS-space (Definition 14 in [5]) when it is a compactly based sober topological space, this is a topological space such that: (DS1) the collection KO(X) forms a basis for the topology τ, (DS2) X, τ is sober. Where in place of Condition (DS2) we could also have:

4 SERGIO A. CELANI AND MARÍA ESTEBAN (DS2 ) X is T 0 and if Z is a closed subset and L is a non-empty down-directed subfamily of KO(X) such that Z U for all U L, then Z {U : U L}. Remark 2.2. DS-spaces were originally defined as ordered topological spaces, where the order considered is nothing more thatn the dual of the specialization order of the space, and this fact simplifyes the definition considerably. We have the following characterization of the distributivity condition for semilattice by means of order ideals and -irreducible meet filters. For a proof of next theorem see Theorem 10 in [5]: Theorem 2.3. Let A be a semilattice. Then A is distributive iff {P c : P Irr (A)} = Id(A). Moreover, for distributive semilattices, we get the following useful characterization of -irreducible meet filters. For a proof see Lemma 6 and Theorem 10 in [5]. Proposition 2.4. Let A be a distributive semilattice and let F Fi(A). following are equivalent: (1) F Irr (A). (2) a, b / F, ( c / F & f F (a f, b f c) ). (3) F c Id(A). The following is a distributive semilattice analogue of Birkhoff s Prime Filter Lemma, and it is proved in [5] (Theorem 8). Lemma 2.5. ( -irreducible Meet Filter Lemma) Let A be a distributive semilattice. Let F Fi(A), and I Id(A) be such that F I =. Then there is G Irr (A) such that F G and G I =. For X = X, τ a given DS-space, consider the family F (X) := {U : U c KO(X)}, which is closed under intersection by Condition (DS1). In [5] (see also [15]) it is proved that F(X) := F (X),, X is a distributive semilattice, called the dual distributive semilattice of X. For A = A,, 1 a given distributive semilatice, consider the map σ A : A P (Irr (A)) defined by σ A (a) = {P Irr (A) : a P }. By convenience, we will omit the subscript of σ A, when no confusion is possible. In [5] it is proved that {σ(a) c : a A} is a basis for a topology τ A on Irr (A). Moreover Irr (A), τ A is shown to be a DS-space, called the dual DS-space of A. If X, τ is a DS-space, then it is homeomorphic to Irr (F(X)), τ F(X) by means of the map ˆε : X Irr (F(X)), given by ˆε (x) = {U F (X) : x U}. If A is a distributive semilatice, then it is isomorphic to F(Irr (A)) by means of the mapping σ. Hilbert algebras. Hilbert algebras represent the algebraic counterpart of the implicative fragment of Intuitionistic Propositional Logic. In this subsection we recall the representation theory for Hilbert algebras. Definition 2.6. A Hilbert algebra is an algebra A = A,, 1 of type (2, 0) such that the following axioms hold in A: (1) a (b a) = 1. (2) (a (b c)) ((a b) (a c)) = 1. The

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM 5 (3) a b = 1 = b a implies a = b. In [10] Diego proves that the class of Hilbert algebras forms a variety. It is easy to see that the binary relation A defined on a Hilbert algebra A by a A b iff a b = 1 is a partial order on A with top element 1. This order is called the natural ordering on A. When the context is clear, we will drop the subscript of A. Let us recall that the class of Hilbert algebras is a subclass of the BCK-algebras (see [9], page 165), more concretely, they are the BCK-algebras that satisfy condition (2) in previous definition. Lemma 2.7. Let A be a Hilbert algebra and a, b, c A. Then the following equalities are satisfied: (1) a a = 1, (2) 1 a = a, (3) a (b c) = b (a c), (4) a (b c) = (a b) (a c), (5) a ((a b) b) = 1, (6) a (a b) = a b, (7) ((a b) b) b = a b, (8) (a b) ((b a) a) = (b a) ((a b) b). An implicative filter (or deductive system) of a Hilbert algebra A is a subset P A such that 1 P and if a, a b P, then b P. Notice that implicative filters are up-sets, and all principal up-sets are implicative filters. We denote by Fi (A) the collection of all implicative filters of A. The set Fi (A) is closed under arbitrary intersections, so for any B A, we denote by B the least implicative filter containing B. We call B the implicative filter generated by B. It is well known that: B = {a A : b 1,..., b n B (b 1 (b 2 (... (b n a)... )) = 1)}. Notice that for all a A, a = a. Consider the bounded lattice Fi (A) := Fi (A),,, A,, where the meet operation is given by intersection, and the join operation is given by the implicative filter generated by the union. We say that an implicative filter P of a Hilbert algebra A is -irreducible when it is a meet irreducible element of the lattice Fi (A), i.e. when for all P 1, P 2 Fi (A), if P = P 1 P 2 then P = P 1 or P = P 2. Since the lattice Fi (A) is distributive (see [10] or [18]), -irreducible implicative filters can be also defined as those implicative filters P for which P 1 P or P 2 P whenever P 1 P 2 P for any P 1, P 2 Fi (A). We denote by Irr (A) the collection of all -irreducible implicative filters of A. In [8] it was studied a representation theorem and a categorical duality for Hilbert algebras using certain ordered sober spaces, that was refined in [7] showing that one can dispense with the order. Following [7], we call dual objects of Hilbert algebras H-spaces. Recall that an H-space is a topological space X = X, τ κ such that: (H1) κ is a basis of open and compact subsets for X, (H2) for every U, V κ, sat(u V c ) κ, (H3) X, τ κ is sober. Where in place of Condition (H3) we could also have: (H3 ) X is T 0 and if Z is a closed subset and L is a non-empty down-directed subfamily of κ such that Z U for all U L, then Z {U : U L}.

6 SERGIO A. CELANI AND MARÍA ESTEBAN For Hilbert algebras, we get the following useful characterization of -irreducible implicative filters. For a proof of the next proposition see [10] or [18]. Proposition 2.8. Let A be a Hilbert algebra. Let P Fi (A). The following are equivalent: (1) P Irr (A). (2) a, b / P, c / P (a c, b c P ). (3) P c Id(A). The following is a Hilbert algebra analogue of Birkhoff s Prime Filter Lemma and it is proved in [3] (Theorem 2.6). Lemma 2.9. ( -Irreducible Implicative Filter Lemma) Let A be an H -algebra, and let P Fi (A), I Id(A) be such that P I =. Then there is Q Irr (A) such that P Q and Q I =. For X = X, τ κ a given H-space, consider the family D(X) := {U : U c κ}, and define a binary operation on it given by U V = (sat(u V c )) c. By Condition (H2) this operation is well defined, and in [7] (see also [8]) it is proved that D(X) := D(X),, X is a Hilbert algebra, called the dual Hilbert algebra of X. For A a given Hilbert algebra, consider the map ϕ A : A P (Irr (A)) defined by ϕ A (a) = {P Irr (A) : a P }. In [8] it is proved that the family κ A := {ϕ A (a) c : a A} is a basis for a topology τ κa on Irr (A). Moreover Irr (A), τ κa is shown to be a H-space, called the dual H-space of A. Moreover, the dual of its specialization order is the inclusion relation, and for any U Irr (A), cl(u) = U. If X, τ is an H-space, then it is homeomorphic to Irr (D(X)), τ κd(x) by means of the map ε X : X Irr (D(X)), given by ε X (x) = {U D(X) : x U}. If A is a Hilbert algebra, then it is isomorphic to D(Irr (A)) by means of the map ϕ A. Two different categories with Hilbert algebras as objects have been considered [8]: one is the usual algebraic category, with algebraic homomorphisms as arrows, and the other one has a weaker notion of morphism, namely, maps that preserve the top element and such that h(a b) h(a) h(b). The later are called semihomomorphisms, and in [8] it is proved that they dually correspond to binary relations R X 1 X 2 between H-spaces, satisfying the following properties: (HR1) R 1 (U) κ 1, for every U κ 2, (HR2) R(x) is a closed subset of X 2, for all x X 1. Such relations are called H-relations. For h : A 1 A 2 a given semi-homomorphism between Hilbert algebras, in [8] it is proved that the relation R h Irr (A 2 ) Irr (A 1 ), given by: (P, Q) R h iff h 1 [P ] Q, is an H-relation between the corresponding dual H-spaces. Moreover, for R X 1 X 2 a given H-relation between H-spaces, the map R : D(X 2 ) D(X 1 ), given by R (U) := {x X 1 : R(x) U} is a semi-homomorphism between the corresponding dual Hilbert algebras. Homomorphisms between Hilbert algebras dually correspond to H-relations R X 1 X 2 that satisfy: (HF) If (x, y) R, then there exists z X 1 such that x z and R(z) = cl(y). Such relations are called functional H-relations. Notice that here the adjective functional has a different meaning as in the extended Priestley duality, since the minimum of R(x), for R a functional H-relation, may not exist. In [8] it is proved

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM 7 that the correspondence between semi-homomorphisms and H-relations restricts to one between homomorphisms and functional H-relations. Hilbert algebras with infimum. Now we are left to define the class of algebras we provide a representation theorem and a categorical duality for, namely the class of Hilbert algebras where the natural order is a semilattice. Definition 2.10. An algebra A = A,,, 1 of type (2, 2, 0) is a Hilbert algebra with infimum or H -algebra if: (1) A,, 1 is a Hilbert algebra, (2) A,, 1 is a meet semilattice with top element 1, (3) for all a, b A, a b = 1 iff a b = a. Notice that Condition (3) in previous definition sets down that the order defined by the implication and the order given by the semilattice coincide. In [11] it is proved that the class of H -algebras is a variety 1 : Theorem 2.11. Let A = A,,, 1 be an algebra of type (2, 2, 0). Then A is an H -algebra if and only if for all a, b, c A: (1) A,, 1 is a Hilbert algebra, (2) A,, 1 is a meet-semilattice with top element 1, (3) a (a b) = a b, (4) (a (b c)) ((a b) (a c)) = 1. Example 2.12. In every meet-semilattice A,, 1 with top element 1 it is possible to define a structure of Hilbert algebra with infimum, considering the implication defined by the order, namely, a b = 1 if a b, and a b = b otherwise. Example 2.13. Implicative semilattices [9], also called Hertz algebras or Brouwerian semilattices, are Hilbert algebras with infimum where moreover, the implication is the right residuum of the meet operation, or equivalently, where the following equation holds: (PA) a (b (a b)) = 1. The following example shows that the class of implicative semilattices is strictly included in the class of H -algebras: Example 2.14. Let A be the distributive meet-semilattice with top element A,, 1 (see Figure 1), where A = {0, a, b, 1} {c i : i ω}. Consider on A the implication defined by the order, i.e. the binary operation on A, such that x y = 1 if x y and x y = y otherwise. Then we have that A := A,,, 1 is a DH -algebra. Moreover a (b (a b)) = 0 1, so A does not satisfy Condition (PA), and thereofore it is not an implicative semilattice. We can see also that does not preserve meets in the second coordinate, since 0 = a (a b) (a a) (a b) = b. It follows from the definition of Hilbert algebra, that the implication operation of a DH -algebra is order preserving in the second coordinate and order reversing in the first coordinate. It should be clear that the implication operation of a DH - algebra does not need to have a left adjoint: take, for example, the implication 1 We note that this result also follows by the results given by P. M. Idziak in [16] for BCKalgebras with lattice operations.

8 SERGIO A. CELANI AND MARÍA ESTEBAN Figure 1. Figure 1 in [2] 1 c 1 c 2 a 0 b defined by the order on the four-element Boolean algebra (viewed as a semilattice). Then we get a DH -algebra whose underlying semilattice is a complete lattice, but the implication does not preserve meets in the second coordinate, so it has no left adjoint. In [12] it is identified a logic H associated with the class of Hilbert algebras with infimum in the following sense: for all Γ {ϕ} F m 2 : Γ H ϕ iff H -algebra A, h Hom(Fm, A), ( h[γ] {1} implies h(ϕ) = 1 ). The logic H extends the implicative fragment of the intuitionistic logic with the connective, and is characterized by the following axioms and rules: (K) ϕ (ψ ϕ) (F) (ϕ (ψ δ)) ((ϕ ψ) (ϕ δ)) (E r) (ϕ ψ) ψ (E l) (ϕ ψ) ϕ (MP) ϕ, ϕ ψ ψ (AB) ϕ ψ, ϕ δ ϕ (ψ δ) We focus our interest in H -algebras where the underlying semilattice is distributive. Definition 2.15. We say that an H -algebra A = A,,, 1 is distributive (or a DH -algebra) when the underlying semilattice A,, 1 is distributive. From Example 2.12 it follows that the class of DH -algebras is strictly included in the class of H -algebras, since on any non-distributive semilattice we may define an implication so an H -algebra that is not distributive is obtained. It is well known that the same way the lattice reduct of a Heyting algebra is distributive, the semilattice reduct of an implicative semilattice is distributive, so the class of implicative semilattices is included in the class of DH -algebras. Notice that it follows from Example 2.14 that this inclusion is strict. Let us conclude this subsection pointing out in a lemma a useful property of H -algebras, that will be used later on: Lemma 2.16. Let A be an H -algebra. Then for all a, b, c A, a (b c) (a b) c. 2 F m denotes the collection of all formulas in the language considered, Fm denotes the H - algebra of formulas and Hom(Fm, A) denotes the collection of homomorphisms from Fm to A.

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM 9 Proof. Let a, b, c A. From a b a we get a (b c) (a b) (b c). From a b b we get b c (a b) c, and so (a b) (b c) (a b) ((a b) c) = (a b) c, and we are done. 3. Filters in H -algebras In an H -algebra A = A,,, 1 we can distinguish two types of filters. On the one hand, we have the collection of implicative filters Fi (A), associated with the (, 1)-reduct of A. On the other hand, we have Fi(A), the collection of meet filters associated with the (, 1)-reduct of A. Both classes of filters play an important role in the representation of H -algebras. In the present section we study the relations between these classes, and besides them, one more notion of filter is considered (see Definition 3.4). Let A be an H -algebra. From Lemma 2.16 it follows that for all B A, B B, and when B A is a finite subset, then B = B. It is easy to prove that all meet filters of A are also implicative filters: let a, a b F Fi(A), then a b = a (a b) F so a b b F, and clearly 1 F, since it is a non-empty up-set. Therefore, we have: Fi(A) Fi (A). Moreover, the following relation between -irreducible implicative filters and - irreducible meet filters holds for any H -algebra: Proposition 3.1. Let A be an H -algebra, then Irr (A) Fi(A) Irr (A). Proof. Let F Irr (A) Fi(A) and let F 1, F 2 Fi(A) be such that F 1 F 2 = F. Since F 1, F 2, F are implicative filters, and F is -irreducible, we get F 1 = F or F 2 = F, therefore F is -irreducible meet filter. The following proposition characterizes the distributivity condition on H -algebras by means of -irreducible implicative filters and -irreducible meet filters: Proposition 3.2. An H -algebra A is distributive iff Irr (A) Irr (A). Proof. Assume that A is distributive and let P Irr (A). On the one hand we have that P Fi (A). On the other hand, by Proposition 2.4, P c Id(A). Then by Proposition 2.8, we conclude that P Irr (A). Let now A be an H -algebra such that Irr (A) Irr (A). Since the (, 1)- reduct of A is a Hilbert algebra, then by assumption and Proposition 2.8 we obtain that P c is an order ideal for all P Irr (A). Thus by Theorem 2.3, the (, 1)- reduct of A is a distributive semilattice, so A is a DH -algebra, as required. The inclusion in Proposition 3.2 may be strict, as the following example shows: Example 3.3. Consider the DH -algebra given in Example 2.14. Let us denote by F ab the set ({a, b}) = {a, b, 1} {c i : i ω}, that is an implicative filter, and by F c the set {c i : i ω} {1}, that is a meet filter. It is easy to see that Fi(A) is the collection of all principal up-sets together with F c. Moreover, Fi (A) is Fi(A) together with F ab. It is not difficult to check that F ab Irr (A), but since it is not closed under meet, F ab / Irr (A). Hence, we have: Irr (A) Irr (A). Sumarizing, for any H -algebra A, not necessarily distributive, we have:

10 SERGIO A. CELANI AND MARÍA ESTEBAN Fi(A) Fi (A) Irr (A) Irr (A) and when A is a DH -algebra, these relations turn to be: Fi(A) Fi (A) Irr (A) Irr (A) Finally we mention one more notion of filter for H -algebras, that was introduced in [11], and correspond to the notion of logical filter for the logic H. Although these filters do not play any role in the representation of DH -algebras, we will obtain a dual characterization of them in the last section of the paper. Definition 3.4. Let A be an H -algebra. We say that H A is an absorbent filter of A if for all a, b A: H Fi (A), if b H, then a (a b) H. We denote by Ab(A) the collection of all absorbent filters of A. We have that a subset P A is an absorbent filter iff P is an H -filter. According to the theory of Abstract Algebraic Logic, a subset P A is a H -filter if for every Γ {ϕ} F m and for every h Hom(Fm, A), if Γ H ϕ, and h[γ] P, then h(ϕ) P. It is easy to prove that Ab(A) Fi(A): let a, b P Ab(A). Clearly P is an up-set and moreover a (a b) P. Since P is an implicative filter, then a b P. Notice that Ab(A) is closed under arbitrary intersections, so, for B A we may consider the least absorbent filter containing B. Unlike the case of meet filters or implicative filters, we do not have an alternative characterization of the absorbent filter generated by a set. But we have the following proposition, that will be used later on: Proposition 3.5. For all F Fi(A), F Ab(A) iff for all a A, F a is a meet filter. Proof. Let F Ab(A) and let a A. If a F there is nothing to prove, so suppose a / F. We claim that F a Fi(A): It is just left to show that F a is closed under meets, so let b, c F a. As F, we may assume that there are b 0,..., b n, c 0,..., c m F a such that b 0 (... (b n b)... ) = 1 and c 0 (... (c m c)... ) = 1. By Lemma 2.16, this implies (b 0 b n ) b = 1 and (c 0 c m ) c = 1. Then we have (b 0 b n c 0 c m ) (b c). Since b 0,..., b n, c 0,..., c m F a and F and a are both closed under meets, then we have d 1 F and d 2 a such that (b 0 b n c 0 c m ) = d 1 d 2 b c. Moreover, by definition of absorbent filter, d 2 (d 1 d 2 ) F F a. And since d 2 a F a, by definition of implicative filter we obtain d 1 d 2 F a, as required For the converse, let F Fi(A) be such that for all a A, F a is a meet filter. We show that F is absorbent. Let b F and a A. We show that a (a b) F : notice first that F a = F {a}. Then, as a a and b F we get by

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM11 hypothesis that a b F a. Now we use the definition of implicative filter generated, and we get that there are c 0,..., c n F, for some n ω, such that c 0 (c 1 (... (c n (a (a b)))... )) = 1. But this implies a (a b) F, as required. 4. Representation theorem for DH -algebras In this section we will define the spectral-like dual of a DH -algebra, called DH -Spectral space, and we will prove that any DH -algebra can be represented by means of a DH -Spectral space. This result is based on the representations of Hilbert algebras and distributive semilattices reviewed in Section 2, and gives the representation of implicative semilattices in [4] as a particular case. Definition 4.1. A DH -Spectral space is a structure X = X, ˆX, τ κ such that: X, τ κ is an H-space, ˆX X generates a sober subspace, and for every {U, V } W κ, with W : (DH 1) U c = cl(u c ˆX), (DH 2) cl(u c V c ˆX) c κ, (DH 3) if cl( {W c : W W} ˆX) U c, then there are W 0,..., W n W, for some n ω, such that cl(w c 0 W c n ˆX) U c. Remark 4.2. We have to be careful when talking about complements, since we are working with two spaces at the same time. We establish here on the following convention: complements are always referred to the set X. By spectral-like duality for Hilbert algebras [7, 8] we get that for any DH - Spectral space X = X, ˆX, τ κ, D(X),, X is a Hilbert algebra. Recall that D(X) = {U : U c κ} and the binary operation is defined as U V = (sat(u V c )) c for all U, V D(X). Moreover, by sobriety of X, τ κ it follows that the space is T 0, and so its specialization pre-order is a partial order. We will deal with its dual order, that we denote by X, or by when the context is clear. Then we have that for all U X, cl(u) = U and sat(u) = U with respect to the order X. Moreover of closed (open) subsets are up-sets (downsets). And by condition (DH 1) we get X = cl(x ˆX) = cl( ˆX). Now we are left to define an operation on D(X) representing the meet operation. The following proposition will be useful for this purpose. Notice that by definition of subspace generated, we know that {U ˆX : U κ} is a basis for the subspace of X generated by ˆX. Proposition 4.3. Let X, ˆX, τ κ be a DH -Spectral space. Then KO( ˆX) = {U ˆX : U κ}. Proof. First we show the inclusion from left to right: let W KO( ˆX). If W = we are done, so assume W. By definition of subspace generated, we get W = {V ˆX : V V} for some set V κ. Since W is compact, there are V 0,..., V n V, for some n ω, such that W = (V 0 ˆX) (V n ˆX) = (V0 c Vn c ) c ˆX. Notice that for each i n, Vi c is closed, so cl(v0 c Vn c ˆX) Vi c for each i n. We obtain that cl(v0 c Vn c ˆX) V0 c Vn c, and then cl(v0 c Vn c ˆX) ˆX V0 c Vn c ˆX. Clearly the reverse inclusion also holds, so we

12 SERGIO A. CELANI AND MARÍA ESTEBAN get (V0 c Vn c ) c ˆX = (cl(v0 c Vn c ˆX)) c ˆX. By condition (DH 2), V := cl(v0 c Vn c ˆX) c κ. Therefore, W = V ˆX for V κ, as required. Now we show the reverse inclusion. We just have to show compactness of U ˆX in ˆX for any U κ. Let U κ and consider W := U ˆX. If W = we are done, so suppose that W = {V ˆX : V V} for some non-empty V κ. We claim that {V c : V V} ˆX U c : let x {V c : V V} ˆX, i.e. x ˆX and x / V for all V V. Therefore x / {V ˆX : V V} = U ˆX, and since x ˆX, then x U c. As U c is an up-set, it follows from the claim that cl( {V c : V V} ˆX) U c. Then by condition (DH 3), there are V 0,..., V n V for some n ω such that cl(v0 c Vn c ˆX) U c. So U (V0 c Vn c ˆX) c = (V0 c ˆX) c (Vn c ˆX) c. Therefore: W = U ˆX ((V c 0 ˆX) c (V c n ˆX) c ) ˆX = ((V c 0 ˆX) c ˆX) ((V c n ˆX) c ˆX) = (V 0 ˆX) (V n ˆX), and therefore W = (V 0 ˆX) (V n ˆX), so W is compact. Corollary 4.4. Let X, ˆX, τ κ be a DH -Spectral space. Then ˆX is a DS-space. Proof. Recall that {U ˆX : U κ} is a basis for ˆX. By definition ˆX is sober, and it is T 0 since the space X = X, τ κ is T 0. Moreover, by Proposition 4.3 ˆX is compactly based, so we are done. By spectral-like duality for distributive semilattices [5] we get that for any DH - Spectral space X = X, ˆX, τ κ, the structure F ( ˆX),, ˆX is a distributive semilattice, where F ( ˆX) = {U c : U KO( ˆX)} = {U ˆX : U D(X)}. Condition (DH 2) ensures us that we can lift to D(X) the meet operation on F ( ˆX) given by intersection, and come up with a binary operation on D(X), given by: U V = cl(u V ˆX). It is not difficult to see that D(X),, X is isomorphic to F ( ˆX),, ˆX by means of the map γ : D(X) F ( ˆX), given by: γ(u) = U ˆX. Clearly γ is a surjective map such that γ(x) = ˆX, and from condition (DH 1) it follows that it is injective. Moreover, from U, V D(X) being up-sets, we get γ(u V ) = cl(u V ˆX) ˆX = U V ˆX = γ(u) γ(v ). Proposition 4.5. Let X, ˆX, τ κ be a DH -Spectral space. Then for all U, V D(X), U V = X iff U V = U Proof. By definition of, we have that U V = X iff U V, so we show U V iff U V = U. If U V, then using condition (DH 1), we get U V = cl(u V ˆX) = cl(u ˆX) = U. Assume now U = U V = cl(u V ˆX), and let x U. By assumption, there is y U V ˆX such that y x. In particular, y V and since V is an up-set and y x, we obtain x V, as required. Corollary 4.6. Let X, ˆX, τ κ be a DH -Spectral space. Then D(X),,, X is a DH -algebra.

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM13 Given a DH -Spectral space X = X, ˆX, τ κ, the DH -algebra ˆD(X) := D(X),,, X will be called the dual DH -algebra of X. Now we provide a construction that shows that any DH -algebra A is (isomorphic to) the dual DH -algebra of some DH -Spectral space. Recall that by spectral-like duality for Hilbert algebras we get that, for any DH - algebra A, the pair Irr (A), τ κa is the dual H-space of A,, 1, where κ A = {ϕ A (a) c : a A}, and ϕ A : A P (Irr (A)), given by ϕ A (a) = {P Irr (A) : a P }, is an isomorphism between Hilbert algebras A,, 1 and D(X),, X. By convenience, we will omit the subscript of ϕ A, when no confusion is possible. Let us consider the subspace of Irr (A), τ κa generated by Irr (A). As κ A = {ϕ(a) c : a A} is a basis for τ κa, then ˆκ A := {U Irr (A) : U κ A } = {ϕ(a) c Irr (A) : a A} is a basis for the induced topology on Irr (A), that we denote by τˆκa. Notice that for each a A we have ϕ(a) c Irr (A) = σ(a) c = {G Irr (A) : a / G}, thus by spectral-like duality for distributive semilattices, we get that Irr (A), τˆκa is the dual DS-space of A,, 1, and it is, in particular, sober. In the following proposition we prove that conditions (DH 1)-(DH 3) are also satisfied by the structure Irr (A), Irr (A), τ κa. Proposition 4.7. Let A be a DH -algebra and let {a, c} B A, B. Then: (1) cl(ϕ(a) Irr (A)) = ϕ(a). (2) cl(ϕ(a) ϕ(c) Irr (A)) = ϕ(a c). (3) If cl( {ϕ(b) : b B} Irr (A)) ϕ(c), then there are n ω and b 0,..., b n B, such that cl(ϕ(b 0 ) ϕ(b n ) Irr (A)) ϕ(c). Proof. For (1), the inclusion from left to right is immediate, since ϕ(a) is an up-set. Let us show the inclusion from right to left: let P ϕ(a), i.e. a P Irr (A). Then P c is an order ideal such that a / P c. Thus by Lemma 2.5 there exists Q Irr (A), such that a Q P. Hence P (ϕ(a) Irr (A)) = cl(ϕ(a) Irr (A)). (2) follows straightforwardly from (1) and the fact that ϕ(a) ϕ(c) Irr (A) = ϕ(a c) Irr (A). For (3) Assume that cl( {ϕ(b) : b B} Irr (A)) ϕ(c). We claim that c B. Suppose towards a contradiction that c / B. Then by Lemma 2.5 there is G Irr (A) such that B G and c / G. Thus G {ϕ(b) : b B} Irr (A) cl( {ϕ(b) : b B} Irr (A)), And fromthe hypothesis we infer that G ϕ(c), a contradiction If c = 1 then ϕ(c) = Irr (A) and there is nothing to prove. So assume c 1. Now since c B and c 1, there are n ω and b 0,..., b n B such that (b 0 b n ) c = 1, i.e. b 0 b n c. So ϕ(b 0 ) ϕ(b n ) Irr (A) = ϕ(b 0 b n ) Irr (A) ϕ(c), and since ϕ(c) is an up-set, we obtain cl(ϕ(b 0 ) ϕ(b n ) Irr (A)) ϕ(c). Theorem 4.8. Let A = A,,, 1 be a DH -algebra. Then Irr (A), Irr (A), τ κa is a DH -Spectral space and ϕ : A D(Irr (A)) is an isomorphism of DH - algebras.

14 SERGIO A. CELANI AND MARÍA ESTEBAN Proof. That Irr (A), Irr (A), τ κa is a DH -Spectral space follows from previous proposition and spectral-like dualities for Hilbert algebras and distributive semilattices, as was already remarked. It follows also that ϕ is an isomorphism of Hilbert algebras A,, 1 and D(Irr (A)),, Irr (A). Moreover, it follows from the definition and item (2) of Proposition 4.7, that: hence we are done. ϕ(a) ϕ(c) = cl(ϕ(a) ϕ(c) Irr (A)) = ϕ(a c) Given a DH -algebra A = A,,,, the DH -Spectral space Irr(A) := Irr (A), Irr (A), τ κa will be called the dual DH -Spectral-space of A. Recall that given a DH -Spectral space X = X, ˆX, τ κ, by spectral-like duality for Hilbert algebras we get that the map ε X : X Irr ( ˆD(X)), given by ε X (x) = {U D(X) : x U}, is a homeomorphism between H-spaces X, τ κ and Irr ( ˆD(X)), τ κ ˆD(X). Moreover, by spectral-like duality for distributive semilattices we get that the map ˆε ˆX : ˆX Irr (F( ˆX)), given by ˆε ˆX(x) = {U F ( ˆX) : x U} = {U ˆX : x U D(X)} = γ[ε X (x)], is a homeomorphism between DS-spaces ˆX, τˆκ and Irr (F( ˆX)), τ F( ˆX). Theorem 4.9. Let X, ˆX, τ κ be a DH -Spectral space. Then ε X [ ˆX] = Irr ( ˆD(X)). Proof. Notice that γ[ε X [ ˆX]] = ˆε ˆX[ ˆX] = Irr (F( ˆX)) = γ[irr ( ˆD(X))], so since γ is an isomorphism between D(X),, X and F( ˆX), we conclude that ε X [ ˆX] = Irr ( ˆD(X)). 5. Categorical duality We extend the topological representation studied in previous section to a dual equivalence of categories. Following the same approach as in [8], we consider two different categories with DH -algebras as objects. The morphisms we condider are, on the one hand, the usual notion of algebraic homomorphism, and on the other hand, a weaker notion that naturally extends the notion of semi-homomorphism between Hilbert algebras introduced in [4]. We will see that the later are represented by certain binary relations, while dual relations of the former turn out to be functional. Definition 5.1. A meet-semi-homomorphism (or -semi-homomorphism) between DH -algebras A 1 and A 2 is a map h : A 1 A 2 such that for all a, b A 1 : (1) h(1 1 ) = 1 2, (2) h(a 1 b) h(a) 2 h(b), (3) h(a 1 b) = h(a) 2 h(b). If moreover h satisfies h(a) 2 h(b) h(a 1 b), then it is called a meethomomorphism (or -homomorphism). Recall that we call semi-homomorphism to those maps between Hilbert algebras that satisfy conditions (1) and (2) in previous definition. Therefore, -semihomomorphisms between DH -algebras are, in particular, semi-homomorphism between the corresponding, 1 reducts of such algebras.

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM15 Definition 5.2. A DH -Spectral morphism between DH -Spectral spaces X 1, ˆX 1, τ κ1 and X 2, ˆX 2, τ κ2 is a relation R X 1 X 2 that satisfies: R is an H-relation between H-spaces X 1, τ κ1 and X 2, τ κ2, and (DH M) R(x) = cl(r(x) ˆX 2 ) for every x ˆX 1. By spectral-like duality for Hilbert algebras we get that for any DH -Spectral morphism R X 1 X 2 between DH -Spectral spaces X 1, ˆX 1, τ κ1 and X 2, ˆX 2, τ κ2, the function R : D(X 2 ) D(X 1 ), given by: R (U) = {x X 1 : R(x) U} is a semi-homomorphism between Hilbert algebras D(X 2 ), 2, X 2 and D(X 1 ), 1, X 1. We also get (see Example 3.1 in [8]) that for a DH -space X, ˆX, τ κ, the order on X, given by the dual of the specialization order, is a functional H- relation. Notice that for all x ˆX, x = ( x ˆX). Therefore satisfies also condition (DH M), and hence it is a DH -Spectral morphism. Furthermore, it follows easily that = id D(X), and for DH -Spectral morphisms R X 1 X 2 and S X 2 X 3, S R = R S. Proposition 5.3. Let R X 1 X 2 be a DH -Spectral morphism between DH - Spectral spaces X 1 and X 2. Then for all U, V D(X 2 ), R (cl(u V ˆX 2 )) = cl( R (U) R (V ) ˆX 1 ). Proof. First we show the inclusion from left to right. Let x R (cl(u V ˆX 2 )). By condition (DH 1) we know that R (cl(u V ˆX 2 )) = cl( R (cl(u V ˆX 2 )) ˆX 1 ). Then there is y ˆX 1 such that y R (cl(u V ˆX 2 )) and y x. By definition we have R(y) cl(u V ˆX 2 ). From U, V being up-sets, it follows that R(y) U V, i.e. y R (U) R (V ). By assumption x y ˆX 1, x cl( R (U) R (V ) ˆX 1 ), as required. Let us show now the reverse inclusion. Since R (cl(u V ˆX 2 )) is an upset, it is enough to show that R (U) R (V ) ˆX 1 R (cl(u V ˆX 2 )). Let x R (U) R (V ) ˆX 1, i.e. R(x) U V and x ˆX 1. In order to show that R(x) cl(u V ˆX 2 ), let y R(x). By condition (DH M) we know that R(x) = cl(r(x) ˆX 2 ). Thus there is y R(x) ˆX 2 such that y y. By assumption, y U V, so, y U V ˆX 2, and therefore y cl(u V ˆX 2 ). Hence we have proved that R(x) cl(u V ˆX 2 ), i.e. x R (cl(u V ˆX 2 )), as required. Corollary 5.4. Let R X 1 X 2 be a DH -Spectral morphism between DH - Spectral spaces X 1 and X 2. Then R is a -semi-homomorphism between dual DH -algebras of X 2 and X 1. Previous corollary provides the dual -semi-homomorphism of a DH -spectral morphism. Before showing how to define a DH -Spectral morphism from a -semihomomorphism, we consider a different kind of relations between DH -Spectral spaces. Following [8] we call a DH -Spectral morphism functional when it satisfies condition (HF) (see page 6). Next corollary follows straightforwardly from above results and spectral-like duality for Hilbert algebras: Corollary 5.5. Let R X 1 X 2 be a DH -Spectral funtional morphism between DH -Spectral spaces X 1 and X 2. Then R is a -homomorphism between dual DH -algebras of X 2 and X 1.

16 SERGIO A. CELANI AND MARÍA ESTEBAN Recall that by spectral-like duality for Hilbert algebras we get that, for any - semi-homomorphism h : A 1 A 2 between DH -algebras A 1 and A 2, the relation R h Irr (A 1 ) Irr (A 2 ), given by (P, Q) R h iff h 1 [P ] Q is an H-relation between dual H-spaces of A 2, 2, 1 2 and A 1, 1, 1 1. It also follows that for the identity morphism id : A A for an DH -algebra A, R id1 =, and for DH -algebras A 1, A 2, A 3 and -semi-homomorphisms h : A 1 A 2, g : A 2 A 3, R g f = R f R g. Lemma 5.6. Let h : A 1 A 2 be a -semi-homomorphism between DH -algebras A 1 and A 2. Then for any P Fi(A 2 ), h 1 [P ] Fi(A 1 ). Proof. Let us show that h 1 [P ] is a meet filter: let a h 1 [P ] and b A 1 be such that a 1 b. Then a 1 b = 1 1, so 1 2 = h(1 1 ) = h(a 1 b) h(a) 2 h(b). Therefore h(a) 2 h(b), and by assumption h(a) P and P is an up-set, thus h(b) P, so b h 1 [P ]. This shows that h 1 [P ] is an up-set. Let a, b h 1 [P ]. Then h(a), h(b) P, and since P is meet filter and h(a 1 b) = h(a) 2 h(b), we have h(a 1 b) P, and then a 1 b h 1 [P ]. This shows that h 1 [P ] is closed under meets. Proposition 5.7. Let h : A 1 A 2 be a -semi-homomorphism between DH - algebras A 1 and A 2. Then for any P Irr (A 2 ), R h (P ) = cl(r h (P ) Irr (A 1 )). Proof. Since R h is an H-relation, we know that R h (P ) is closed, so cl(r h (P ) Irr (A 1 )) R h (P ). Let us show the reverse inclusion: recall that the closure of a given subset coincides with the up-set generated, so let Q R h (P ), i.e. h 1 [P ] Q Irr (A 1 ). By Lemma 5.6, h 1 [P ] Fi(A 1 ). So we have h 1 [P ] Q c =, for Q c an order ideal and h 1 [P ] a meet filter, and thus by Lemma 2.5 there is Q Irr (A 1 ) such that h 1 [P ] Q and Q Q c =. Then Q is the required element such that Q R h (P ) Irr (A 1 ) and Q Q. Corollary 5.8. Let h : A 1 A 2 be a -semi-homomorphism between DH - algebras A 1 and A 2. Then R h is a DH -Spectral morphism between dual DH - Spectral spaces of A 2 and A 1. Moreover, if h is a -homomorphism, then R h is functional. 5.1. Dual equivalences of categories. We show first that if we take DH - Spectral spaces as objects and DH -Spectral morphisms as morphisms, we obtain indeed a category. As a corollary we get that if we take DH -Spectral spaces as objects and DH -Spectral functional morphisms as morphisms, we obtain a subcategory of the former. Theorem 5.9. Let X 1, ˆX 1, τ κ1, X 2, ˆX 2, τ κ2 and X 3, ˆX 3, τ κ3 be DH -Spectral spaces and let R X 1 X 2 and S X 2 X 3 be DH -Spectral morphisms. Then: (1) The DH -Spectral morphism 2 X 2 X 2 satisfies: 2 R = R and S 2 = S. (2) S R X 1 X 3 is a DH -Spectral morphism. (3) If R, S are functional, then so is S R.

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM17 Proof. (1) This has been proved in Theorem 3.1 in [8] for H-relations, so it holds in particular for DH -Spectral morphisms. (2) By Theorem 3.1 in [8], we get that S R is an H-relation. We just have to show that S R satisfies condition (DH M), i.e. to show that for all x ˆX 1 : S R(x) = cl(s R(x) ˆX 3 ). Let x ˆX 1. First we prove that S R(x) is an up-set: let z S R(x) and let z 3 w for some w X 3. By definition there is y X 2 such that y R(x) and z S(y). By condition (DH M) on R, R(x) = cl(r(x) ˆX 2 ). Then, there is y R(x) ˆX 2 such that y 2 y. Now since S 2 = S, we get z S(y ). And since y ˆX 2, by condition (DH M) on S, S(y ) = cl(s(y ) ˆX 3 ). Then, there is z S(y ) ˆX 3 such that z 3 z 3 w. Therefore, we have w S(y ), and then from y R(x), we get w S R(x). From S R(x) being an up-set, it is immediate that cl(s R(x) ˆX 3 ) S R(x). For the other inclusion, let z S R(x). By a similar argument as before, we conclude that there is z S R(x) ˆX 3 such that z z, therefore z cl(s R(x) ˆX 3 ). (3) This follows from item (2) and results in Section 5.2 in [8]. Corollary 5.10. DH -Spectral spaces and DH -Spectral morphisms form a category. Proof. For a DH -Spectral space X, we already pointed out that the order on X given by the dual of the specialization order is a DH -Spectral morphism. Then by item (1) in Theorem 5.9, it is the identity morphism on X. By item (2) in Theorem 5.9, relational composition works as composition between DH -Spectral morphisms. Corollary 5.11. DH -Spectral spaces and DH -Spectral functional morphisms from a category. Proof. This follows from previous corollary and item (3) in Theorem 5.9. Remark 5.12. Morphisms between DH -Spectral spaces are DH -Spectral morphisms, and we have denoted them as R X 1 X 2, for DH -Spectral spaces X 1 and X 2. In the usual notation on category theory, they are denoted as R : X 1 X 2. Let us employ the notation used so far, and keep in mind this correspondence. Let us consider the following categories: DH S DH H Sp DH M Sp DH F DH -algebras and -semi-homomorphisms, DH -algebras and -homomorphisms, DH -Spectral spaces and DH -Spectral morphisms, DH -Spectral spaces and DH -Spectral functional morphisms. We complete now the dualities by showing the contravariant functors and the natural isomorphisms involved on them. From previous results we can define a contravariant functor () : DH S Sp DH M such that for DH -algebras A, A 1, A 2 and -semi-homomorphism h : A 1 A 2 : A = Irr (A), Irr (A), τ κa

18 SERGIO A. CELANI AND MARÍA ESTEBAN h = R h Similarly, we define the contravariant functor () : Sp DH M DH S such that for DH -Spectral spaces X, X 1, X 2 and DH -Spectral morphism R X 1 X 2 : X = D(X),,, X R = R Consider now the family of morphisms in DH S: Φ = ( ϕ A : A D(Irr (A)) ) A DH S Theorem 5.13. Φ is a natural isomorphism between the identity functor in DH S and (() ). Proof. Let A 1, A 2 be two DH -algebras and let h : A 1 A 2 be a -semihomomorphism between them. By Lemma 3.5 in [8] we get that Rh ϕ A1 = ϕ A2 h and by Theorem 4.8 we get that for all A DH S, ϕ A is an isomorphism. Recall that for any DH -Spectral space X, ˆX, τ κ, we defined the map ε X : X Irr ( ˆD(X)). Associated with this map, we define now the relation E X X Irr ( ˆD(X) as follows: (x, P ) E X iff ε X (x) P. Lemma 5.14. E X is a DH -Spectral functional morphism. Proof. By results in [8] we know that E X is a functional H-relation, so we just have to check that condition (DH M) is satisfied: let x ˆX. It is immediate that cl(e X (x) Irr ( ˆD(X))) E X (x), so we just have to check the other inclusion. Let P E X (x), i.e. ε X (x) P. By Theorem 4.9 we know that ε X (x) Irr ( ˆD(X)) and clearly ε X (x) E X (x). Therefore P cl(e X (x) Irr ( ˆD(X))), as required. Consider now the family of morphisms in Sp DH M : Σ = ( E X X Irr ( ˆD(X)) ). X Sp DH M Theorem 5.15. Σ is a natural isomorphism between then identity functor in Sp DH M and (() ). Proof. Let X 1, X 2 be two DH -Spectral spaces and let R X 1 X 2 be a DH - Spectral morphism between them. By Lemma 3.4 in [8] we get that (x, y) R iff (ε X1 (x), ε X2 (y)) R R, and from this it follows that R R E X1 = E X2 R. Moreover, by Theorem 4.9 we have that ε X is an homeomorphism such that ε X [ ˆX] = Irr ( ˆD(X)). This implies, together with results from Theorem 3.2 in [8], that E X is an isomorphism in Sp DH M, as required. Corollary 5.16. The categories Sp DH M and DH S are dually equivalent by means of the contravariant functors () and () and the natural equivalences Φ and Σ. Similarly, the categories Sp DH F and DH H are dually equivalent.

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM19 5.2. Duality for other classes of algebras. Notice that DH -algebras is not a variety, because distributivity for meet-semilattices is not an equational condition, but we could consider suclasses of DH -algebras that are indeed varieties, such as the class of those DH -algebras where the underlying meet-semilattice is a distributive lattice. It would be interesting to investigate wether a similar strategy than the one used in the present paper could be followed and would provide spectrallike duality for such variety. Another subclass of DH -algebras already mentioned are implicative semilattices (or IS-algebras). A Spectral-like duality for IS-algebras was studied in [4], where IS-spaces are defined as those DS-spaces X, τ that satisfy the following condition: (IS) for all U, V KO(X), sat(u V c ) KO(X). Notice that condition (IS) is similar to condition (H2) with κ = KO(X). Let us call IS-Spectral spaces those DH -Spectral spaces X, ˆX, τ κ where X = ˆX. Notice that in this case and by Proposition 4.3 we have KO(X) = KO( ˆX) = {U ˆX : U κ} = κ. We show now that these spaces are the same as the spectralduals given by the representation of IS-algebras in [4]. Hence we recover the duality studied there, as a particular case of our duality. It is immediate that if X, ˆX, τ κ is an IS-Spectral space, then X, τ κ is an IS-space. For the converse, let X, τ be an IS-space. We show that the structure X, X, τ KO(X) is an IS-Spectral space. By assumption X, τ KO(X) is an H-Spectral space. Clearly X X generates a sober subspace. Moreover, for all {U, V } W KO(X), as U c is closed, it is immediate that cl(u c X) = U c, so condition (DH 1) is satisfied. Also cl(u c V c X) c = U V KO(X), so condition (DH 2) is satisfied. Finally, if cl( {W c : W W} X) = {W c : W W} U c, as U is compact, there are W 0,..., W n W, for some n ω, such that W c 0 W c n = cl(w c 0 W c n X) U c. So condition (DH 3) is also satisfied. Dual morphisms of homomorphisms between IS-spaces are defined as functional DH -Spectral homomorphisms, taking into account that KO(X) plays the roll of κ. Therefore, from the correspondence for objects it follows easily the correspondence for morphisms. We conclude that the duality given in [4] is a particular case of the one presented here. 6. Topological characterization of filters In Section 5 of [8] the authors give a topological characterization of implicative filters of an H-algebra. Here we give a topological characterization of implicative filters, meet filters and absorbent filters of a DH -algebra. Let A = A,,, 1 be a DH -algebra. Recall that we denote the topological space Irr (A), τ κa by Irr (A). Let us define the following maps: C ( ) : Fi (A) C(Irr (A)) F C F = {ϕ(a) : a F } F ( ) : C(Irr (A)) Fi (A) C F C = {a A : C ϕ(a)}