SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT …

SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT

ALGEBRAS WITH INFIMUM

SERGIO A. CELANI AND MARIA ESTEBAN

Abstract. Distributive Hilbert Algebras with infimum, or DH∧-algebras, are

algebras with implication and conjunction that are not the residuum of eachother. 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 topologicalspaces. We prove that the category of these spaces, together with certain kind

of relations, is dually equivalent to the category of DH∧-algebras and ∧-semi-

homomorphisms. We restrict this result to give a duality for the categoryof 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 distributivelattices in at least three ways (cf. [1] and its references). The approach initiated byStone himself [21] leads to a representation in terms of compactly-based sober spacesin which the collection of compact open sets is closed under finite intersections.Further generalizations of this approach lead to dualities for distributive meet-semilattices [5, 15], implicative semilattices [3], Hilbert algebras [8] and Hilbertalgebras with supremum [7]. What they all have in common is that they providerepresentations in terms of compactly-based sober spaces. We refer to this class ofdualities as spectral-like.

A different approach initiated by Priestley [19] leads to a representation in termsof ordered Hausdorff topological spaces. Although both approaches have been fol-lowed to generalize the pioneering work on representation of Boolean algebras withoperators [17], the later has been held to be advantageous [6, 13], specially in viewof recent developments of the theory of canonical extensions (see [14] and his ref-erences). The keypoint of this theory is that the additional n-ary operations eitherpreserve joins (meets) in each coordinate, or send joins (meets) in each coordinate tomeets (joins). Such n-ary operations are represented in terms of n+ 1-ary relationssatisfying certain conditions.

In the present paper we focus our attention on a class of algebras that is notwithin 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 MARIA ESTEBAN

nor reverse joins or meets in each coordinate. The largest class of algebras towhich our study can be applied, is distributive meet-semilattices extended with aHilbert-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 thesealgebras distributive Hilbert algebras with infimum (or DH∧-algebras). The lackof residuation is precisely what forces us to search a completly different route for atopological representation of this class of algebras, that will be suported in alreadyexisting dualities for distributive meet-semilattices and Hilbert algebras. In thepresent 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 opera-tion on the Hilbert algebra, instead of what is customary, namely looking at theimplication as an additional operation on the (semi)-lattice structure. Accordingly,the meet is represented by a subset satisfying certain conditions, instead of by arelation. A parallel study involving a Priestley-style duality for the same class ofalgebras will be developed in a forthcoming paper.

The organization of the paper goes as follows. In Section 2 we present the pre-liminaries and we establish the basic notational conventions. Particularly, we recallthe spectral-like duality for distributive meet-semilattices given in [5] and [15], thespectral-like duality for Hilbert algebras developed in [8] and [7], and we introducethe class of DH∧-algebras. In Section 3 we examine different notions of filters asso-ciated with a DH∧-algebra, and the relations between them, that yield the keypointof our representation strategy. In Section 4 we present the duality for objects, whereduals of DH∧-algebras are certain kind of spectral-like spaces augmented with asubset that satisfies some conditions. In Section 5 we extend the duality for mor-phisms between DH∧-algebras. Following [8], we deal with two different notionsof morphism, namely, the usual algebraic notion of homomorphism and a weakernotion related to that of semi-homomorphism between Hilbert algebras. Then twocategories are defined and the dual equivalences of categories is proved. Finally inSection 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 generatedby 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 denotesthe complement of Y , namely {x ∈ X : x /∈ Y }. The symbol ω denotes the setof all natural numbers. Let L be a lattice. Recall that an element a ∈ L is meetirreducible when for all b, c ∈ L, if a = b ∧ c then a = b or a = c. Similarly, anelement a is meet prime when for all b, c ∈ L, if b ∧ c ≤ a then b ≤ c or c ≤ a. Itis known that when L is a distributive lattice, then prime an irreducible elementscoincide.

Let X1, X2 be two sets and let R ⊆ X1×X2 be a binary relation relation betweenthem. For any x1 ∈ X1, R(x1) = {x2 ∈ X2 : (x1, x2) ∈ R}, and for each Y ⊆ X2,R−1(Y ) = {x1 ∈ X1 : ∃y ∈ Y ((x1, y) ∈ R)}. For given sets X1, X2, X3, functionsf : X1 −→ X2 , g : X2 −→ X3 and relations R ⊆ X1 ×X2 and S ⊆ X2 ×X3, thecomposition 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 referto it simply by X. For each Y ⊆ X, cl(Y ) denotes the closure of Y . Recall that asubset Y ⊆ X is saturated when it is an intersection of open sets. The saturation ofa 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 iffx ∈ 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 = Y1 ∪ Y2 implies Y = Y1 or Y = Y2 forany closed subsets Y1, Y2 ∈ C(X). The space X is sober if each closed irreduciblesubset is the closure of a unique point.

Distributive meet-semilattices. Let us recall that a meet-semilattice with topelement 1, is an algebra A = 〈A,∧, 1〉 of type (2, 0) such that the operation ∧ isidempotent, commutative, associative, and a ∧ 1 = a, for all a ∈ A. As usual, thebinary relation ≤ defined by a ≤ b iff a ∧ b = a is a partial order. In what followswe 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 6= ∅ 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 principaldown-sets are, in particular, order ideals.

A meet filter of a semilattice A is a non-empty up-set closed under the meetoperation, 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 isproper when F 6= 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 denoteby JB〉〉 the least meet filter containing B. We call JB〉〉 the meet filter generatedby B. It is well known that JB〉〉 = {a ∈ A : ∃b1, . . . , bn ∈ B (b1 ∧ · · · ∧ bn ≤ a)}.Notice that for all a ∈ A, Ja〉〉 = ↑a. We consider the bounded lattice Fi(A) :=〈Fi(A),∩,∨, A, ∅〉, where the meet operation is given by intersection and the joinoperation 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 meetirreducible 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 thata ∧ 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 fromStone’s work in [21], or more detailed in [15], where Gratzer considers distribu-tive semilattices as a general framework to discuss topological representations ofdistributive lattices [15]. He studies some elementary properties distributive semi-lattices, 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 stud-ied in [5], where dual objects are topological spaces called DS-spaces. Recall thatX = 〈X, τ〉 is a DS-space (Definition 14 in [5]) when it is a compactly based sobertopological 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:


(DS2’) X is T0 and if Z is a closed subset and L is a non-empty down-directedsubfamily of KO(X) such that Z ∩ U 6= ∅ for all U ∈ L, then Z ∩

⋂{U :

U ∈ L} 6= ∅.

Remark 2.2. DS-spaces were originally defined as ordered topological spaces, wherethe order considered is nothing more thatn the dual of the specialization order ofthe space, and this fact simplifyes the definition considerably.

We have the following characterization of the distributivity condition for semi-lattice by means of order ideals and ∧-irreducible meet filters. For a proof of nexttheorem 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 characteri-zation of ∧-irreducible meet filters. For a proof see Lemma 6 and Theorem 10in [5].

Proposition 2.4. Let A be a distributive semilattice and let F ∈ Fi(A). Thefollowing 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 FilterLemma, 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 thedual distributive semilattice of X.

For A = 〈A,∧, 1〉 a given distributive semilatice, consider the map σA : A −→P↑(Irr∧(A)) defined by σA(a) = {P ∈ Irr∧(A) : a ∈ P}. By convenience, we willomit the subscript of σA, when no confusion is possible. In [5] it is proved that{σ(a)c : a ∈ A} is a basis for a topology τA on Irr∧(A). Moreover 〈Irr∧(A), τA〉 isshown 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)), τF(X)〉 by means of the map ε : X →Irr∧(F(X)), given by ε (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 im-plicative fragment of Intuitionistic Propositional Logic. In this subsection we recallthe representation theory for Hilbert algebras.

Definition 2.6. A Hilbert algebra is an algebra A = 〈A,→, 1〉 of type (2, 0) suchthat the following axioms hold in A:

(1) a→ (b→ a) = 1.(2) (a→ (b→ c))→ ((a→ b)→ (a→ c)) = 1.


(3) a→ b = 1 = b→ a implies a = b.

In [10] Diego proves that the class of Hilbert algebras forms a variety. It is easyto see that the binary relation ≤A defined on a Hilbert algebra A by a ≤A b iffa→ b = 1 is a partial order on A with top element 1. This order is called the naturalordering on A. When the context is clear, we will drop the subscript of ≤A. Let usrecall 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 equal-ities 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 subsetP ⊆ A such that 1 ∈ P and if a, a → b ∈ P , then b ∈ P . Notice that implicativefilters are up-sets, and all principal up-sets are implicative filters. We denote byFi→(A) the collection of all implicative filters of A. The set Fi→(A) is closed underarbitrary intersections, so for any B ⊆ A, we denote by 〈B〉 the least implicativefilter containing B. We call 〈B〉 the implicative filter generated by B. It is wellknown that: 〈B〉 = {a ∈ A : ∃b1, . . . , bn ∈ B (b1 → (b2 → (. . . (bn → 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 thejoin 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 whenit is a meet irreducible element of the lattice Fi→(A), i.e. when for all P1, P2 ∈Fi→(A), if P = P1 ∩ P2 then P = P1 or P = P2. Since the lattice Fi→(A) isdistributive (see [10] or [18]), →-irreducible implicative filters can be also definedas those implicative filters P for which P1 ⊆ P or P2 ⊆ P whenever P1 ∩ P2 ⊆ Pfor any P1, P2 ∈ Fi→(A). We denote by Irr→(A) the collection of all→-irreducibleimplicative filters of A.

In [8] it was studied a representation theorem and a categorical duality for Hilbertalgebras using certain ordered sober spaces, that was refined in [7] showing that onecan dispense with the order. Following [7], we call dual objects of Hilbert algebrasH-spaces. Recall that an H-space is a topological space X = 〈X, τκ〉 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, τκ〉 is sober.

Where in place of Condition (H3) we could also have:

(H3’) X is T0 and if Z is a closed subset and L is a non-empty down-directedsubfamily of κ such that Z ∩ U 6= ∅ for all U ∈ L, then Z ∩

⋂{U : U ∈ L} 6=

∅.


For Hilbert algebras, we get the following useful characterization of→-irreducibleimplicative 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 areequivalent:

(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 Lemmaand 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, τκ〉 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. ByCondition (H2) this operation is well defined, and in [7] (see also [8]) it is provedthat D(X) := 〈D(X),⇒, X〉 is a Hilbert algebra, called the dual Hilbert algebra ofX.

For A a given Hilbert algebra, consider the map ϕA : A −→ P↑(Irr→(A)) definedby ϕA(a) = {P ∈ Irr→(A) : a ∈ P}. In [8] it is proved that the family κA :={ϕA(a)c : a ∈ A} is a basis for a topology τκA

on Irr→(A). Moreover 〈Irr→(A), τκA〉

is shown to be a H-space, called the dual H-space of A. Moreover, the dual of itsspecialization 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)), τκD(X)〉 by

means of the map εX : X → Irr→(D(X)), given by εX (x) = {U ∈ D(X) : x ∈ U}.If A is a Hilbert algebra, then it is isomorphic to D(Irr→(A)) by means of the mapϕA.

Two different categories with Hilbert algebras as objects have been considered [8]:one is the usual algebraic category, with algebraic homomorphisms as arrows, andthe other one has a weaker notion of morphism, namely, maps that preserve thetop element and such that h(a → b) ≤ h(a) → h(b). The later are called semi-homomorphisms, and in [8] it is proved that they dually correspond to binaryrelations R ⊆ X1 ×X2 between H-spaces, satisfying the following properties:

(HR1) R−1(U) ∈ κ1, for every U ∈ κ2,(HR2) R(x) is a closed subset of X2, for all x ∈ X1.

Such relations are calledH-relations. For h : A1 −→ A2 a given semi-homomorphismbetween Hilbert algebras, in [8] it is proved that the relation Rh ⊆ Irr→(A2) ×Irr→(A1), given by: (P,Q) ∈ Rh iff h−1[P ] ⊆ Q, is an H-relation between the cor-responding dual H-spaces. Moreover, for R ⊆ X1×X2 a given H-relation betweenH-spaces, the map �R : D(X2) −→ D(X1), given by �R(U) := {x ∈ X1 : R(x) ⊆U} is a semi-homomorphism between the corresponding dual Hilbert algebras.

Homomorphisms between Hilbert algebras dually correspond to H-relations R ⊆X1 ×X2 that satisfy:

(HF) If (x, y) ∈ R, then there exists z ∈ X1 such that x ≤ z and R(z) = cl(y).

Such relations are called functional H-relations. Notice that here the adjectivefunctional has a different meaning as in the extended Priestley duality, since theminimum of R(x), for R a functional H-relation, may not exist. In [8] it is proved


that the correspondence between semi-homomorphisms and H-relations restricts toone between homomorphisms and functional H-relations.

Hilbert algebras with infimum. Now we are left to define the class of algebraswe provide a representation theorem and a categorical duality for, namely the classof 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 algebrawith 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 definedby the implication and the order given by the semilattice coincide. In [11] it isproved that the class of H∧-algebras is a variety1:

Theorem 2.11. Let A = 〈A,→,∧, 1〉 be an algebra of type (2, 2, 0). Then A is anH∧-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 possibleto 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 Brouwe-rian semilattices, are Hilbert algebras with infimum where moreover, the implicationis the right residuum of the meet operation, or equivalently, where the followingequation holds:

(PA) a→ (b→ (a ∧ b)) = 1.

The following example shows that the class of implicative semilattices is strictlyincluded 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} ∪ {ci : i ∈ ω}. Consider on A the implicationdefined by the order, i.e. the binary operation→ on A, such that x→ y = 1 if x ≤ yand x→ y = y otherwise. Then we have that A := 〈A,→,∧, 1〉 is a DH∧-algebra.Moreover a → (b → (a ∧ b)) = 0 6= 1, so A does not satisfy Condition (PA), andthereofore it is not an implicative semilattice. We can see also that→ does not pre-serve meets in the second coordinate, since 0 = a→ (a∧b) 6= (a→ a)∧(a→ b) = b.

It follows from the definition of Hilbert algebra, that the implication operationof a DH∧-algebra is order preserving in the second coordinate and order reversingin 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

1We note that this result also follows by the results given by P. M. Idziak in [16] for BCK-algebras with lattice operations.


Figure 1. Figure 1 in [2]

0

a b

c1

c2

1

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, butthe implication does not preserve meets in the second coordinate, so it has no leftadjoint.

In [12] it is identified a logic H∧ associated with the class of Hilbert algebraswith infimum in the following sense: for all Γ ∪ {ϕ} ⊆ Fm2:

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 theconnective ∧, 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 dis-

tributive.

Definition 2.15. We say that an H∧-algebra A = 〈A,→,∧, 1〉 is distributive (ora DH∧-algebra) when the underlying semilattice 〈A,∧, 1〉 is distributive.

From Example 2.12 it follows that the class of DH∧-algebras is strictly includedin the class of H∧-algebras, since on any non-distributive semilattice we may definean implication so an H∧-algebra that is not distributive is obtained. It is wellknown 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 ofimplicative semilattices is included in the class of DH∧-algebras. Notice that itfollows from Example 2.14 that this inclusion is strict.

Let us conclude this subsection pointing out in a lemma a useful property ofH∧-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.

2Fm 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.


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 theone 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 filtersassociated with the (∧, 1)-reduct of A. Both classes of filters play an important rolein the representation of H∧-algebras. In the present section we study the relationsbetween these classes, and besides them, one more notion of filter is considered (seeDefinition 3.4).

Let A be an H∧-algebra. From Lemma 2.16 it follows that for all B ⊆ A,〈B〉 ⊆ JB〉〉, and when B ⊆ A is a finite subset, then JB〉〉 = ↑

∧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 anon-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 F1, F2 ∈ Fi(A) be such that F1 ∩ F2 = F .Since F1, F2, F are implicative filters, and F is →-irreducible, we get F1 = F orF2 = F , therefore F is ∧-irreducible meet filter. �

The following proposition characterizes the distributivity condition onH∧-algebrasby 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 wehave that P ∈ Fi→(A). On the other hand, by Proposition 2.4, P c ∈ Id(A). Thenby 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 obtainthat 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 denoteby Fab the set ↑({a, b}) = {a, b, 1} ∪ {ci : i ∈ ω}, that is an implicative filter, andby Fc the set {ci : i ∈ ω} ∪ {1}, that is a meet filter. It is easy to see that Fi(A) isthe collection of all principal up-sets together with Fc. Moreover, Fi→(A) is Fi(A)together with Fab. It is not difficult to check that Fab ∈ Irr→(A), but since it isnot closed under meet, Fab /∈ Irr∧(A) . Hence, we have: Irr∧(A) ( Irr→(A).

Sumarizing, for any H∧-algebra A, not necessarily distributive, we have:


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 forH∧-algebras, that was introducedin [11], and correspond to the notion of logical filter for the logic H∧. Althoughthese filters do not play any role in the representation of DH∧-algebras, we willobtain 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 absorbentfilter 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 asubset P ⊆ A is an absorbent filter iff P is an H∧-filter. According to the theory ofAbstract Algebraic Logic, a subset P ⊆ A is a H∧-filter if for every Γ∪ {ϕ} ⊆ Fmand 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 isan up-set and moreover a → (a ∧ b) ∈ P . Since P is an implicative filter, thena ∧ b ∈ P .

Notice that Ab(A) is closed under arbitrary intersections, so, for B ⊆ A we mayconsider the least absorbent filter containing B. Unlike the case of meet filters orimplicative filters, we do not have an alternative characterization of the absorbentfilter generated by a set. But we have the following proposition, that will be usedlater on:

Proposition 3.5. For all F ∈ Fi(A), F ∈ Ab(A) iff for all a ∈ A, 〈F ∪ ↑a〉 is ameet filter.

Proof. Let F ∈ Ab(A) and let a ∈ A. If a ∈ F there is nothing to prove, sosuppose 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 6= ∅, we may assumethat there are b0, . . . , bn, c0, . . . , cm ∈ F ∪ ↑a such that b0 → (. . . (bn → b) . . . ) = 1and c0 → (. . . (cm → c) . . . ) = 1. By Lemma 2.16, this implies (b0∧· · ·∧bn)→ b = 1and (c0 ∧ · · · ∧ cm)→ c = 1. Then we have (b0 ∧ · · · ∧ bn ∧ c0 ∧ · · · ∧ cm) ≤ (b ∧ c).Since b0, . . . , bn, c0, . . . , cm ∈ F ∪↑a and F and ↑a are both closed under meets, thenwe have d1 ∈ F and d2 ∈ ↑a such that (b0∧· · ·∧ bn∧ c0∧· · ·∧ cm) = d1∧d2 ≤ b∧ c.Moreover, by definition of absorbent filter, d2 → (d1∧d2) ∈ F ⊆ 〈F∪↑a〉. And sinced2 ∈ ↑a ⊆ 〈F ∪ ↑a〉, by definition of implicative filter we obtain d1 ∧ d2 ∈ 〈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 filtergenerated, and we get that there are c0, . . . , cn ∈ F , for some n ∈ ω, such thatc0 → (c1 → (. . . (cn → (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, calledDH∧-Spectral space, and we will prove that any DH∧-algebra can be representedby means of a DH∧-Spectral space. This result is based on the representations ofHilbert algebras and distributive semilattices reviewed in Section 2, and gives therepresentation of implicative semilattices in [4] as a particular case.

Definition 4.1. A DH∧-Spectral space is a structure X = 〈X, X, τκ〉 such that:

• 〈X, τκ〉 is an H-space,

• X ⊆ X generates a sober subspace,and for every {U, V } ∪W ⊆ κ, with W 6= ∅:

(DH∧1) U c = cl(U c ∩ X),

(DH∧2) cl(U c ∩ V c ∩ X)c ∈ κ,

(DH∧3) if cl(⋂{W c : W ∈ W} ∩ X) ⊆ U c, then there are W0, . . . ,Wn ∈ W,

for some n ∈ ω, such that cl(W c0 ∩ · · · ∩W c

n ∩ X) ⊆ U c.

Remark 4.2. We have to be careful when talking about complements, since we areworking with two spaces at the same time. We establish here on the followingconvention: 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, X, τκ〉, 〈D(X),⇒, X〉 is a Hilbert algebra. Recall thatD(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, τκ〉 it follows thatthe space is T0, and so its specialization pre-order is a partial order. We will dealwith its dual order, that we denote by ≤X , or by ≤ when the context is clear. Thenwe 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 ∩ X) = cl(X).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 definitionof subspace generated, we know that {U ∩ X : U ∈ κ} is a basis for the subspace

of X generated by X.

Proposition 4.3. Let 〈X, X, τκ〉 be a DH∧-Spectral space. Then KO(X) = {U ∩X : U ∈ κ}.

Proof. First we show the inclusion from left to right: let W ∈ KO(X). If W = ∅ weare done, so assumeW 6= ∅. By definition of subspace generated, we getW =

⋃{V ∩

X : V ∈ V} for some set V ⊆ κ. Since W is compact, there are V0, . . . , Vn ∈ V,

for some n ∈ ω, such that W = (V0 ∩ X) ∪ · · · ∪ (Vn ∩ X) = (V c0 ∩ · · · ∩ V cn )c ∩ X.

Notice that for each i ≤ n, V ci is closed, so cl(V c0 ∩ · · · ∩ V cn ∩ X) ⊆ V ci for each

i ≤ n. We obtain that cl(V c0 ∩ · · · ∩V cn ∩ X) ⊆ V c0 ∩ · · · ∩V cn , and then cl(V c0 ∩ · · · ∩V cn ∩ X) ∩ X ⊆ V c0 ∩ · · · ∩ V cn ∩ X. Clearly the reverse inclusion also holds, so we


get (V c0 ∩ · · · ∩ V cn )c ∩ X = (cl(V c0 ∩ · · · ∩ V cn ∩ X))c ∩ X. By condition (DH∧2),

V := cl(V c0 ∩ · · · ∩ V cn ∩ X)c ∈ κ. Therefore, W = V ∩ X for V ∈ κ, as required.

Now we show the reverse inclusion. We just have to show compactness of U ∩ Xin X for any U ∈ κ. Let U ∈ κ and consider W := U ∩X. If W = ∅ we are done, sosuppose that W =

⋃{V ∩ X : V ∈ V} for some non-empty V ⊆ κ. We claim that⋂

{V c : V ∈ V}∩ X ⊆ U c: let x ∈⋂{V c : V ∈ V}∩ X, i.e. x ∈ X and x /∈ V for all

V ∈ V. Therefore x /∈⋃{V ∩ X : V ∈ V} = U ∩ X, and since x ∈ X, then x ∈ U c.

As U c is an up-set, it follows from the claim that cl(⋂{V c : V ∈ V} ∩ X) ⊆ U c.

Then by condition (DH∧3), there are V0, . . . , Vn ∈ V for some n ∈ ω such that

cl(V c0 ∩· · ·∩V cn∩X) ⊆ U c. So U ⊆ (V c0 ∩ · · · ∩ V cn∩X)c = (V c0 ∩X)c∪· · ·∪(V cn∩X)c.Therefore:

W = U ∩ X ⊆ ((V c0 ∩ X)c ∪ · · · ∪ (V cn ∩ X)c) ∩ X

= ((V c0 ∩ X)c ∩ X) ∪ · · · ∪ ((V cn ∩ X)c ∩ X) = (V0 ∩ X) ∪ · · · ∪ (Vn ∩ X),

and therefore W = (V0 ∩ X) ∪ · · · ∪ (Vn ∩ X), so W is compact. �

Corollary 4.4. Let 〈X, X, τκ〉 be a DH∧-Spectral space. Then X is a DS-space .

Proof. Recall that {U ∩ X : U ∈ κ} is a basis for X. By definition X is sober,

and it is T0 since the space X = 〈X, τκ〉 is T0. Moreover, by Proposition 4.3 X iscompactly based, so we are done. �

By spectral-like duality for distributive semilattices [5] we get that for any DH∧-

Spectral space X = 〈X, X, τκ〉, the structure 〈F (X),∩, X〉 is a distributive semi-

lattice, where F (X) = {U c : U ∈ KO(X)} = {U ∩ X : U ∈ D(X)}. Condition

(DH∧2) ensures us that we can lift to D(X) the meet operation on F (X) given byintersection, and come up with a binary operation u on D(X), given by:

U u V = cl(U ∩ V ∩ X).

It is not difficult to see that 〈D(X),u, X〉 is isomorphic to 〈F (X),∩, X〉 by

means of the map γ : D(X) −→ F (X), given by:

γ(U) = U ∩ X.

Clearly γ is a surjective map such that γ(X) = X, and from condition (DH∧1)it follows that it is injective. Moreover, from U, V ∈ D(X) being up-sets, we get

γ(U u V ) = cl(U ∩ V ∩ X) ∩ X = U ∩ V ∩ X = γ(U) ∩ γ(V ).

Proposition 4.5. Let 〈X, X, τκ〉 be a DH∧-Spectral space. Then for all U, V ∈D(X), U ⇒ V = X iff U u V = U

Proof. By definition of ⇒, we have that U ⇒ V = X iff U ⊆ V , so we showU ⊆ V iff U u V = U . If U ⊆ V , then using condition (DH∧1), we get U u V =

cl(U ∩ V ∩ X) = cl(U ∩ X) = U . Assume now U = U u V = cl(U ∩ V ∩ X), and

let x ∈ U . By assumption, there is y ∈ U ∩ V ∩ X 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, X, τκ〉 be a DH∧-Spectral space. Then 〈D(X),⇒,u, X〉 isa DH∧-algebra.


Given a DH∧-Spectral space X = 〈X, X, τκ〉, the DH∧-algebra

D(X) := 〈D(X),⇒,u, X〉

will be called the dual DH∧-algebra of X. Now we provide a construction thatshows that any DH∧-algebra A is (isomorphic to) the dual DH∧-algebra of someDH∧-Spectral space.

Recall that by spectral-like duality for Hilbert algebras we get that, for anyDH∧-algebra A, the pair 〈Irr→(A), τκA

〉 is the dual H-space of 〈A,→, 1〉, where κA ={ϕA(a)c : a ∈ A}, and ϕA : A −→ P↑(Irr→(A)), given by ϕA(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 ϕA, when no confusion is possible.

Let us consider the subspace of 〈Irr→(A), τκA〉 generated by Irr∧(A). As κA =

{ϕ(a)c : a ∈ A} is a basis for τκA, then

κA := {U ∩ Irr∧(A) : U ∈ κA} = {ϕ(a)c ∩ Irr∧(A) : a ∈ A}

is a basis for the induced topology on Irr∧(A), that we denote by τκA. 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), τκA〉

is the dual DS-space of 〈A,∧, 1〉, and it is, in particular, sober. In the followingproposition we prove that conditions (DH∧1)-(DH∧3) are also satisfied by thestructure 〈Irr→(A), Irr∧(A), τκA

〉.

Proposition 4.7. Let A be a DH∧-algebra and let {a, c} ∪B ⊆ A, B 6= ∅. 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

b0, . . . , bn ∈ B, such that cl(ϕ(b0) ∩ · · · ∩ ϕ(bn) ∩ 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 ∈ JB〉〉. Suppose towards a contradiction that c /∈ JB〉〉. Then by Lemma 2.5there is G ∈ Irr∧(A) such that JB〉〉 ⊆ 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 contradictionIf c = 1 then ϕ(c) = Irr→(A) and there is nothing to prove. So assume c 6= 1.

Now since c ∈ JB〉〉 and c 6= 1, there are n ∈ ω and b0, . . . , bn ∈ B such that(b0 ∧ · · · ∧ bn) → c = 1, i.e. b0 ∧ · · · ∧ bn ≤ c. So ϕ(b0) ∩ · · · ∩ ϕ(bn) ∩ Irr∧(A) =ϕ(b0 ∧ · · · ∧ bn) ∩ Irr∧(A) ⊆ ϕ(c), and since ϕ(c) is an up-set, we obtain cl(ϕ(b0) ∩· · · ∩ ϕ(bn) ∩ Irr∧(A)) ⊆ ϕ(c). �

Theorem 4.8. Let A = 〈A,→,∧, 1〉 be a DH∧-algebra. Then 〈Irr→(A), Irr∧(A), τκA〉

is a DH∧-Spectral space and ϕ : A −→ D(Irr→(A)) is an isomorphism of DH∧-algebras.


Proof. That 〈Irr→(A), Irr∧(A), τκA〉 is a DH∧-Spectral space follows from previous

proposition and spectral-like dualities for Hilbert algebras and distributive semilat-tices, as was already remarked. It follows also that ϕ is an isomorphism of Hilbertalgebras 〈A,→, 1〉 and 〈D(Irr→(A)),⇒, Irr→(A)〉. Moreover, it follows from thedefinition and item (2) of Proposition 4.7, that:

ϕ(a) u ϕ(c) = cl(ϕ(a) ∩ ϕ(c) ∩ Irr∧(A)) = ϕ(a ∧ c)

hence we are done. �

Given a DH∧-algebra A = 〈A,→,∧, 〉, the DH∧-Spectral space Irr(A) :=〈Irr→(A), Irr∧(A), τκA

〉 will be called the dual DH∧-Spectral-space of A. Re-

call that given a DH∧-Spectral space X = 〈X, X, τκ〉, by spectral-like duality

for Hilbert algebras we get that the map εX : X −→ Irr→(D(X)), given byεX(x) = {U ∈ D(X) : x ∈ U}, is a homeomorphism between H-spaces 〈X, τκ〉and 〈Irr→(D(X)), τκD(X)

〉. Moreover, by spectral-like duality for distributive semi-

lattices we get that the map εX : X −→ Irr∧(F(X)), given by

εX(x) = {U ∈ F (X) : x ∈ U} = {U ∩ X : x ∈ U ∈ D(X)} = γ[εX(x)],

is a homeomorphism between DS-spaces 〈X, τκ〉 and 〈Irr∧(F(X)), τF(X)〉.

Theorem 4.9. Let 〈X, X, τκ〉 be a DH∧-Spectral space. Then εX [X] = Irr∧(D(X)).

Proof. Notice that γ[εX [X]] = εX [X] = Irr∧(F(X)) = γ[Irr∧(D(X))], so since γ

is an isomorphism between 〈D(X),u, X〉 and F(X), we conclude that εX [X] =

Irr∧(D(X)). �

5. Categorical duality

We extend the topological representation studied in previous section to a dualequivalence of categories. Following the same approach as in [8], we consider twodifferent categories with DH∧-algebras as objects. The morphisms we condider are,on the one hand, the usual notion of algebraic homomorphism, and on the otherhand, a weaker notion that naturally extends the notion of semi-homomorphismbetween Hilbert algebras introduced in [4]. We will see that the later are repre-sented by certain binary relations, while dual relations of the former turn out to befunctional.

Definition 5.1. A meet-semi-homomorphism (or ∧-semi-homomorphism) betweenDH∧-algebras A1 and A2 is a map h : A1 −→ A2 such that for all a, b ∈ A1:

(1) h(11) = 12,(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 meet-homomorphism (or ∧-homomorphism).

Recall that we call semi-homomorphism to those maps between Hilbert alge-bras that satisfy conditions (1) and (2) in previous definition. Therefore, ∧-semi-homomorphisms between DH∧-algebras are, in particular, semi-homomorphismbetween the corresponding 〈→, 1〉 reducts of such algebras.


Definition 5.2. ADH∧-Spectral morphism betweenDH∧-Spectral spaces 〈X1, X1, τκ1〉

and 〈X2, X2, τκ2〉 is a relation R ⊆ X1 ×X2 that satisfies:

• R is an H-relation between H-spaces 〈X1, τκ1〉 and 〈X2, τκ2

〉, and

(DH∧M) R(x) = cl(R(x) ∩ X2) for every x ∈ X1.

By spectral-like duality for Hilbert algebras we get that for any DH∧-SpectralmorphismR ⊆ X1×X2 betweenDH∧-Spectral spaces 〈X1, X1, τκ1

〉 and 〈X2, X2, τκ2〉,

the function �R : D(X2) −→ D(X1), given by:

�R(U) = {x ∈ X1 : R(x) ⊆ U}is a semi-homomorphism between Hilbert algebras 〈D(X2),⇒2, X2〉 and 〈D(X1),⇒1

, X1〉. We also get (see Example 3.1 in [8]) that for a DH∧-space 〈X, X, τκ〉, theorder ≤ on X, given by the dual of the specialization order, is a functional H-relation. Notice that for all x ∈ X, ↑x = ↑(↑x ∩ X). Therefore ≤ satisfies alsocondition (DH∧M), and hence it is a DH∧-Spectral morphism. Furthermore, itfollows easily that �≤ = idD(X), and for DH∧-Spectral morphisms R ⊆ X1 ×X2

and S ⊆ X2 ×X3, �S◦R = �R ◦�S .

Proposition 5.3. Let R ⊆ X1 ×X2 be a DH∧-Spectral morphism between DH∧-Spectral spaces X1 and X2. Then for all U, V ∈ D(X2),

�R(cl(U ∩ V ∩ X2)) = cl(�R(U) ∩�R(V ) ∩ X1).

Proof. First we show the inclusion from left to right. Let x ∈ �R(cl(U∩V ∩X2)). By

condition (DH∧1) we know that �R(cl(U∩V ∩X2)) = cl(�R(cl(U∩V ∩X2))∩X1).

Then there is y ∈ X1 such that y ∈ �R(cl(U ∩V ∩X2)) and y ≤ x. By definition we

have R(y) ⊆ cl(U ∩V ∩X2). From U, V being up-sets, it follows that R(y) ⊆ U ∩V ,

i.e. y ∈ �R(U)∩�R(V ). By assumption x ≥ y ∈ X1, x ∈ cl(�R(U)∩�R(V )∩X1),as required.

Let us show now the reverse inclusion. Since �R(cl(U ∩ V ∩ X2)) is an up-

set, it is enough to show that �R(U) ∩ �R(V ) ∩ X1 ⊆ �R(cl(U ∩ V ∩ X2)). Let

x ∈ �R(U) ∩ �R(V ) ∩ X1, i.e. R(x) ⊆ U ∩ V and x ∈ X1. In order to show

that R(x) ⊆ cl(U ∩ V ∩ X2), let y ∈ R(x). By condition (DH∧M) we know that

R(x) = cl(R(x) ∩ X2). Thus there is y′ ∈ R(x) ∩ X2 such that y′ ≤ y. By

assumption, y′ ∈ U ∩ V , so, y′ ∈ U ∩ V ∩ X2, and therefore y ∈ cl(U ∩ V ∩ X2).

Hence we have proved that R(x) ⊆ cl(U ∩ V ∩ X2), i.e. x ∈ �R(cl(U ∩ V ∩ X2)),as required. �

Corollary 5.4. Let R ⊆ X1 ×X2 be a DH∧-Spectral morphism between DH∧-Spectral spaces X1 and X2. Then �R is a ∧-semi-homomorphism between dualDH∧-algebras of X2 and X1.

Previous corollary provides the dual ∧-semi-homomorphism of a DH∧-spectralmorphism. Before showing how to define a DH∧-Spectral morphism from a ∧-semi-homomorphism, we consider a different kind of relations between DH∧-Spectralspaces. Following [8] we call a DH∧-Spectral morphism functional when it satisfiescondition (HF) (see page 6). Next corollary follows straightforwardly from aboveresults and spectral-like duality for Hilbert algebras:

Corollary 5.5. Let R ⊆ X1 ×X2 be a DH∧-Spectral funtional morphism betweenDH∧-Spectral spaces X1 and X2. Then �R is a ∧-homomorphism between dualDH∧-algebras of X2 and X1.


Recall that by spectral-like duality for Hilbert algebras we get that, for any ∧-semi-homomorphism h : A1 −→ A2 between DH∧-algebras A1 and A2, the relationRh ⊆ Irr→(A1)× Irr→(A2), given by

(P,Q) ∈ Rh iff h−1[P ] ⊆ Q

is an H-relation between dual H-spaces of 〈A2,→2, 12〉 and 〈A1,→1, 11〉. It alsofollows that for the identity morphism id : A −→ A for an DH∧-algebra A, Rid1

=⊆, and for DH∧-algebras A1,A2,A3 and ∧-semi-homomorphisms h : A1 −→ A2,g : A2 −→ A3, Rg◦f = Rf ◦Rg.

Lemma 5.6. Let h : A1 −→ A2 be a ∧-semi-homomorphism between DH∧-algebrasA1 and A2. Then for any P ∈ Fi(A2), h−1[P ] ∈ Fi(A1).

Proof. Let us show that h−1[P ] is a meet filter: let a ∈ h−1[P ] and b ∈ A1 be suchthat a ≤1 b. Then a →1 b = 11, so 12 = h(11) = 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, thush(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), wehave h(a ∧1 b) ∈ P , and then a ∧1 b ∈ h−1[P ]. This shows that h−1[P ] is closedunder meets. �

Proposition 5.7. Let h : A1 −→ A2 be a ∧-semi-homomorphism between DH∧-algebras A1 and A2. Then for any P ∈ Irr∧(A2), Rh(P ) = cl(Rh(P ) ∩ Irr∧(A1)).

Proof. Since Rh is an H-relation, we know that Rh(P ) is closed, so cl(Rh(P ) ∩Irr∧(A1)) ⊆ Rh(P ). Let us show the reverse inclusion: recall that the closure of agiven subset coincides with the up-set generated, so let Q ∈ Rh(P ), i.e. h−1[P ] ⊆Q ∈ Irr→(A1). By Lemma 5.6, h−1[P ] ∈ Fi(A1). So we have h−1[P ] ∩ Qc = ∅,for Qc an order ideal and h−1[P ] a meet filter, and thus by Lemma 2.5 there isQ′ ∈ Irr∧(A1) such that h−1[P ] ⊆ Q′ and Q′ ∩ Qc = ∅. Then Q′ is the requiredelement such that Q′ ∈ Rh(P ) ∩ Irr∧(A1) and Q′ ⊆ Q. �

Corollary 5.8. Let h : A1 −→ A2 be a ∧-semi-homomorphism between DH∧-algebras A1 and A2. Then Rh is a DH∧-Spectral morphism between dual DH∧-Spectral spaces of A2 and A1. Moreover, if h is a ∧-homomorphism, then Rh isfunctional.

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 obtainindeed a category. As a corollary we get that if we take DH∧-Spectral spacesas objects and DH∧-Spectral functional morphisms as morphisms, we obtain asubcategory of the former.

Theorem 5.9. Let 〈X1, X1, τκ1〉, 〈X2, X2, τκ2

〉 and 〈X3, X3, τκ3〉 be DH∧-Spectral

spaces and let R ⊆ X1×X2 and S ⊆ X2×X3 be DH∧-Spectral morphisms. Then:

(1) The DH∧-Spectral morphism ≤2⊆ X2 ×X2 satisfies:

≤2 ◦R = R and S ◦ ≤2= S.

(2) S ◦R ⊆ X1 ×X3 is a DH∧-Spectral morphism.(3) If R,S are functional, then so is S ◦R.


Proof. (1) This has been proved in Theorem 3.1 in [8] for H-relations, so it holdsin 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 ∈ X1:

S ◦R(x) = cl(S ◦R(x) ∩ X3).

Let x ∈ X1. First we prove that S ◦R(x) is an up-set: let z ∈ S ◦R(x) andlet z ≤3 w for some w ∈ X3. By definition there is y ∈ X2 such that y ∈ R(x)

and z ∈ S(y). By condition (DH∧M) on R, R(x) = cl(R(x) ∩ X2). Then, there is

y′ ∈ R(x) ∩ X2 such that y′ ≤2 y. Now since S ◦ ≤2= S, we get z ∈ S(y′). And

since y′ ∈ X2, by condition (DH∧M) on S, S(y′) = cl(S(y′) ∩ X3). Then, there is

z′ ∈ S(y′) ∩ X3 such that z′ ≤3 z ≤3 w. Therefore, we have w ∈ S(y′), and thenfrom 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)∩X3) ⊆ 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)∩X3 such that z′ ≤ z, therefore z ∈ cl(S ◦R(x)∩X3).

(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 cat-egory.

Proof. For a DH∧-Spectral space X, we already pointed out that the order ≤ on Xgiven by the dual of the specialization order is a DH∧-Spectral morphism. Thenby item (1) in Theorem 5.9, it is the identity morphism on X. By item (2) inTheorem 5.9, relational composition works as composition between DH∧-Spectralmorphisms. �

Corollary 5.11. DH∧-Spectral spaces and DH∧-Spectral functional morphismsfrom 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 mor-phisms, and we have denoted them asR ⊆ X1×X2, forDH∧-Spectral spaces X1 andX2. In the usual notation on category theory, they are denoted as R : X1 −→ X2.Let us employ the notation used so far, and keep in mind this correspondence.

Let us consider the following categories:

DH∧S DH∧-algebras and ∧-semi-homomorphisms,DH∧H DH∧-algebras and ∧-homomorphisms,

SpDH∧

M DH∧-Spectral spaces and DH∧-Spectral morphisms,

SpDH∧

F DH∧-Spectral spaces and DH∧-Spectral functional morphisms.

We complete now the dualities by showing the contravariant functors and thenatural isomorphisms involved on them. From previous results we can define a

contravariant functor ()∗ : DH∧S −→ SpDH∧

M such that for DH∧-algebras A,A1,A2

and ∧-semi-homomorphism h : A1 −→ A2:

A∗ = 〈Irr→(A), Irr∧(A), τκA〉


h∗ = Rh

Similarly, we define the contravariant functor ()∗ : SpDH∧

M −→ DH∧S such thatfor DH∧-Spectral spaces X,X1,X2 and DH∧-Spectral morphism R ⊆ X1 ×X2:

X∗ = 〈D(X),⇒,u, X〉

R∗ = �R

Consider now the family of morphisms in DH∧S :

Φ =(ϕA : A −→ D(Irr→(A))

)A∈DH∧

S

Theorem 5.13. Φ is a natural isomorphism between the identity functor in DH∧Sand (()∗)

∗.

Proof. Let A1,A2 be two DH∧-algebras and let h : A1 −→ A2 be a ∧-semi-homomorphism between them. By Lemma 3.5 in [8] we get that �Rh

◦ ϕA1=

ϕA2◦ h and by Theorem 4.8 we get that for all A ∈ DH∧S , ϕA is an isomorphism. �

Recall that for any DH∧-Spectral space 〈X, X, τκ〉, we defined the map εX :

X −→ Irr→(D(X)). Associated with this map, we define now the relation EX ⊆X × Irr→(D(X) as follows:

(x, P ) ∈ EX iff εX(x) ⊆ P.

Lemma 5.14. EX is a DH∧-Spectral functional morphism.

Proof. By results in [8] we know that EX is a functional H-relation, so we just

have to check that condition (DH∧M) is satisfied: let x ∈ X. It is immediate that

cl(EX(x)∩ Irr∧(D(X))) ⊆ EX(x), so we just have to check the other inclusion. Let

P ∈ EX(x), i.e. εX(x) ⊆ P . By Theorem 4.9 we know that εX(x) ∈ Irr∧(D(X))

and clearly εX(x) ∈ EX(x). Therefore P ∈ cl(EX(x) ∩ Irr∧(D(X))), as required.�

Consider now the family of morphisms in SpDH∧

M :

Σ =(EX ⊆ X × Irr→(D(X))

)X∈SpDH∧

M

.

Theorem 5.15. Σ is a natural isomorphism between then identity functor in

SpDH∧

M and (()∗)∗.

Proof. Let X1,X2 be two DH∧-Spectral spaces and let R ⊆ X1 ×X2 be a DH∧-Spectral morphism between them. By Lemma 3.4 in [8] we get that (x, y) ∈ Riff (εX1

(x), εX2(y)) ∈ R�R

, and from this it follows that R�R◦ EX1

= EX2◦

R. Moreover, by Theorem 4.9 we have that εX is an homeomorphism such thatεX [X] = Irr∧(D(X)). This implies, together with results from Theorem 3.2 in [8],

that EX is an isomorphism in SpDH∧

M , as required. �

Corollary 5.16. The categories SpDH∧

M and DH∧S are dually equivalent by meansof the contravariant functors ()∗ and ()∗ and the natural equivalences Φ and Σ.

Similarly, the categories SpDH∧

F and DH∧H are dually equivalent.


5.2. Duality for other classes of algebras. Notice that DH∧-algebras is not avariety, because distributivity for meet-semilattices is not an equational condition,but we could consider suclasses of DH∧-algebras that are indeed varieties, suchas the class of those DH∧-algebras where the underlying meet-semilattice is adistributive lattice. It would be interesting to investigate wether a similar strategythan the one used in the present paper could be followed and would provide spectral-like duality for such variety.

Another subclass of DH∧-algebras already mentioned are implicative semilat-tices (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 followingcondition:

(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, X, τκ〉 where X =

X. Notice that in this case and by Proposition 4.3 we have KO(X) = KO(X) =

{U ∩ X : U ∈ κ} = κ. We show now that these spaces are the same as the spectral-duals 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, X, τκ〉is an IS-Spectral space, then 〈X, τκ〉 is an IS-space.

For the converse, let 〈X, τ〉 be an IS-space. We show that the structure 〈X,X, τKO(X)〉is an IS-Spectral space. By assumption 〈X, τKO(X)〉 is anH-Spectral space. ClearlyX ⊆ 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

W0, . . . ,Wn ∈ W, for some n ∈ ω, such that W c0∩· · ·∩W c

n = cl(W c0∩· · ·∩W c

n∩X) ⊆U c. So condition (DH∧3) is also satisfied.

Dual morphisms of homomorphisms between IS-spaces are defined as functionalDH∧-Spectral homomorphisms, taking into account that KO(X) plays the roll of κ.Therefore, from the correspondence for objects it follows easily the correspondencefor morphisms. We conclude that the duality given in [4] is a particular case of theone presented here.

6. Topological characterization of filters

In Section 5 of [8] the authors give a topological characterization of implicativefilters of an H-algebra. Here we give a topological characterization of implicativefilters, meet filters and absorbent filters of a DH∧-algebra.

Let A = 〈A,→,∧, 1〉 be a DH∧-algebra. Recall that we denote the topologicalspace 〈Irr→(A), τκA

〉 by Irr→(A). Let us define the following maps:

C(−) : Fi→(A) −→ C(Irr→(A))

F 7−→ CF =⋂{ϕ(a) : a ∈ F}

F(−) : C(Irr→(A)) −→ Fi→(A)

C 7−→ FC = {a ∈ A : C ⊆ ϕ(a)}


By Proposition 5.1 in [8] we get that these maps are well defined, and moreover,they are inverses of each other, and order isomorphisms between ordered spaces〈Fi→(A),⊆〉 and 〈C(Irr→(A)),⊇〉. Therefore, implicative filters of A correspondto closed subsets of 〈Irr→(A), τκA

〉 and viceversa. Moreover, as the space Irr→(A) issober, it follows that→-irreducible implicative filters of A correspond to irreducibleclosed subsets of Irr→(A). We identify now which closed subsets of 〈Irr→(A), τκA

〉correspond with meet filters of A:

Proposition 6.1. For any DH∧-algebra A:

(1) If F ∈ Fi(A), then CF = cl(CF ∩ Irr∧(A)).(2) If C ∈ C(Irr→(A)) is such that C = cl(C ∩ Irr∧(A)), then FC ∈ Fi(A).

Proof. (1) For F ∈ Fi(A), as CF is closed then cl(CF ∩ Irr∧(A)) ⊆ CF . We showthe other inclusion: let P ∈ Irr→(A) be such that P ∈ CF . Then F ⊆ P . Considerthe order ideal P c. We have F ∩ P c = ∅, so by Lemma2.5, there is F ′ ∈ Irr∧(A)such that F ⊆ F ′ and F ′ ∩ P c = ∅, i.e. F ′ ⊆ P . Thus F ′ ∈ CF ∩ Irr∧(A) andtherefore P ∈ cl(CF ∩ Irr∧(A)).

(2) For C ∈ C(Irr→(A)) a closed subset such that C = cl(C ∩ Irr∧(A)), leta, b ∈ FC . We show a∧b ∈ FC . By assumption, C ⊆ ϕ(a), ϕ(b). Then C∩Irr∧(A) ⊆ϕ(a)∩ϕ(b)∩ Irr∧(A), and then C = cl(C ∩ Irr∧(A)) ⊆ cl(ϕ(a)∩ϕ(b)∩ Irr∧(A)) =ϕ(a ∧ b). Therefore a ∧ b ∈ FC . �

Recall that we have already considered the topological space 〈Irr∧(A), τκA〉.

By definition we have that C ∈ C(Irr∧(A)) iff C = C ′ ∩ Irr∧(A) for some C ′ ∈C(Irr→(A)). Therefore, by means of the same maps as before we obtain that meetfilters of A correspond to closed sets of 〈Irr∧(A), τκA

〉 and viceversa. Moreover, asthe subspace generated by Irr∧(A) is also sober, we have that ∧-irreducible meetfilters correspond to irreducible closed subsets of 〈Irr∧(A), τκA

〉.Finally we identify now which closed sets of 〈Irr→(A), τκA

〉 correspond withabsorbent filters of A:

Proposition 6.2. For any DH∧-algebra A:

(1) If F ∈ Ab(A), then for all a ∈ A, CF ∩ ϕ(a) = cl(CF ∩ ϕ(a) ∩ Irr∧(A)).(2) If C ∈ C(Irr→(A)) is such that C ∩ ϕ(a) = cl(C ∩ ϕ(a) ∩ Irr∧(A)) for all

a ∈ A, then FC ∈ Ab(A).

Proof. (1) For F ∈ Ab(A) and a ∈ A, if a ∈ F then CF∩ϕ(a) = CF , and since F is ameet filter, using Proposition 6.1 we obtain CF ∩ϕ(a) = CF = cl(CF ∩ Irr∧(A)) =cl(CF ∩ ϕ(a) ∩ Irr∧(A)). Assume that a /∈ F . Then by Proposition 3.5 we getthat 〈F ∪ ↑a〉 is a meet filter. Let P ∈ CF ∩ ϕ(a), i.e. {a} ∪ F ⊆ P . We showP ∈ cl(CF ∩ϕ(a)∩ Irr∧(A)). By previous claim, we have a meet filter 〈F ∪ ↑a〉 andan order ideal P c such that 〈F ∪ ↑a〉∩P c = ∅. By Lemma 2.5 there is F ′ ∈ Irr∧(A)such that F ′ ∩ P c = ∅ and 〈F ∪ ↑a〉 ⊆ F ′, i.e. {a} ∪ F ⊆ F ′ ⊆ P for F ′ ∈ Irr∧(A).Therefore F ′ ∈ CF ∩ ϕ(a) ∩ Irr∧(A), and moreover P ∈ cl(CF ∩ ϕ(a) ∩ Irr∧(A)),as required.

(2) For C ∈ C(Irr→(A)) a closed subset such that C ∩ ϕ(a) = cl(C ∩ ϕ(a) ∩Irr∧(A)) for all a ∈ A. We show that FC is an absorbent filter, i.e. we show thatfor any b ∈ FC and c ∈ A, c → (b ∧ c) ∈ FC . By definition, we have to show thatC ⊆ ϕ(c→ (b∧c)) = ϕ(c)⇒ (ϕ(b) u ϕ(c)). By assumption C∩ϕ(c) ⊆ cl(C∩ϕ(c)∩Irr∧(A)) and C ⊆ ϕ(b). Then C ∩ ϕ(c) ⊆ cl(ϕ(b) ∩ ϕ(c) ∩ Irr∧(A)) = ϕ(b) u ϕ(c),so C ∩ ϕ(c) ∩ (ϕ(b) u ϕ(c))c = ∅. Since C is closed, it is an up-set, so this implies


C ∩ sat(ϕ(c) ∩ (ϕ(b) u ϕ(c))c) = ∅, i.e. C ⊆ (sat(ϕ(c) ∩ (ϕ(b) u ϕ(c))c))c = ϕ(a →(b ∧ a)), as required. �

Therefore, by means of the same maps as before, we obtain that absorbent filtersof A correspond to closed sets C of 〈Irr∧(A), τκA

〉 with the property that for alla ∈ A, C ∩ ϕ(a) = cl(C ∩ ϕ(a) ∩ Irr∧(A)).

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CONICET and Departamento de Matematicas, Facultad de Ciencias Exactas. Univ.

Nac. del Centro. Pinto 399, 7000 Tandil, ArgentinaE-mail address: [email protected]

Departament de Logica, Historia i Filosofia de la Ciencia, Facultat de Filosofia,

Universitat de Barcelona (UB). Montalegre 6, 08001 Barcelona, SpainE-mail address: [email protected]

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