SIMPLE AND SUBDIRECTLY IRREDUCIBLES BOUNDED DISTRIBUTIVE LATTICES WITH UNARY OPERATORS

Distributive lattices with operators (DLO) are a natural generalization of the notion of Boolean algebras with operators. An operator in a bounded distributive lattice A is a function f : An→ A which preserves ∧ (or ∨) in each coordinate. In the last few years these classes of algebras have been actively investigated since they appear as algebraic counterpart of many logics. Some important contributions in this area have been the papers of Goldblatt [12], Petrovich [16], and Sofronie-Stokkermans [18] which deal with the representation and topological duality for DLO. More recently, in [11] Gehrke et al. have studied conditions for canonicity and an automatic mechanism for the translation of equations that are Sahlqvist. In [17] Sofronie-Stokkermans studies a uniform presentation of representation and decibility results related to a Kripke-style semantics, and the link between algebraic and Kripke-style semantics of several nonclassical logics. Positive modal logic was introduced by Dunn in [10], and it corresponds to the positive fragment of the local modal consequence relation defined by the class of all Kripke frames. The algebraic semantic of this fragment is the variety of positive modal algebras (or PM-algebras) introduced in [10], and further studied by means of topological methods in [7], and in [6] by methods from abstract algebraic logic. A PM-algebra is a bounded distributive lattice with two unary modal operators and ♦ satisfying additional conditions that relate to these operators. Topological Boolean algebras or closure Boolean algebras were given by McKinsey and Tarski [15] to conduct an algebraic study of topological spaces (see also [4]). In [13],


Introduction
Distributive lattices with operators (DLO) are a natural generalization of the notion of Boolean algebras with operators.An operator in a bounded distributive lattice A is a function f : A n → A which preserves ∧ (or ∨) in each coordinate.
In the last few years these classes of algebras have been actively investigated since they appear as algebraic counterpart of many logics.Some important contributions in this area have been the papers of Goldblatt [12], Petrovich [16], and Sofronie-Stokkermans [18] which deal with the representation and topological duality for DLO.More recently, in [11] Gehrke et al. have studied conditions for canonicity and an automatic mechanism for the translation of equations that are Sahlqvist.In [17] Sofronie-Stokkermans studies a uniform presentation of representation and decibility results related to a Kripke-style semantics, and the link between algebraic and Kripke-style semantics of several nonclassical logics.
Positive modal logic was introduced by Dunn in [10], and it corresponds to the positive fragment of the local modal consequence relation defined by the class of all Kripke frames.The algebraic semantic of this fragment is the variety of positive modal algebras (or PM-algebras) introduced in [10], and further studied by means of topological methods in [7], and in [6] by methods from abstract algebraic logic.A PM-algebra is a bounded distributive lattice with two unary modal operators and ♦ satisfying additional conditions that relate to these operators.
Topological Boolean algebras or closure Boolean algebras were given by McKinsey and Tarski [15] to conduct an algebraic study of topological spaces (see also [4]).In [13], Halmos introduced monadic Boolean algebras for an algebraic study of the one variable fragment of predicate logic.Important classes of bounded distributive lattices with operators that generalize the monadic Boolean algebras are the Q-distributive lattices introduced by Cignoli [9], and the monadic Heyting algebras studied by Bezhanishvili [2].
In research on DLO, Priestley duality is a useful tool.The dual space associated with a DLO is a Priestley space with a (n + 1)-ary relation for each n-ary operator (see [12,18]).The dual spaces of PM-algebras can be defined with a unique binary relation or with two binary relations, but strongly related between them (see [7]).In Birchall's Master's thesis [3] there is a duality for perfect distributive lattices with unary operators, and there are some results on subdirect irreducibility using topological duality.
Since any variety of algebras is determinated by the subdirectly irreducible algebras, it is important to have a characterization of them in varieties of (DLO).The main contribution of this paper aims at giving a characterization of simple and subdirectly irreducible of certain varieties of positive modal algebras, and certain varieties of Heyting algebras with unary operators.
The paper is structured as follows: in Section 2 we start by recalling some basic definitions and results on the Priestley duality for distributive lattices with unary operators.This duality has been developed in [12,18] for the general case, in [16] for the case of distributive lattices with an operator of type ♦, and in [7] for positive modal algebras.In Section 3, we will give a topological characterization of congruences and we will determine the simple and subdirectly irreducible algebras for some varieties of distributive lattices with unary operators of type and ♦.In Section 4 we will introduce the variety of topological positive modal algebras (TPM-algebras) and the variety of monadic positive modal algebras (MPM-algebras) as a generalization of the closure Boolean algebras and monadic Boolean algebras, respectively.The TPM-algebras and MPM-algebras can be considered as the algebraic semantics of the positive fragment of the local modal consequence relation defined by the class of all Kripke frames X,R , where the binary relation R is reflexive and transitive, and R is an equivalence, respectively.It is interesting to note that the characteristic axiom (a ∨ b) ≤ a ∨ ♦b, valid in positive modal algebras, is not true in monadic Heyting algebras, thus the →-free reduct of a monadic Heyting algebra is not an MPM-algebra.In Section 5 we will study the simple and subdirectly irreducible algebras in some varieties of Heyting algebras with modal operators and ♦ ( ♦-Heyting algebras).Some related results on Heyting algebras with operators appear in [14] (see also [19]).We will prove that every -congruence in ♦-Heyting algebras is also a ♦congruence, but there exists ♦-congruences that are not -congruences.This shows a certain asymmetry between the modal operators and ♦ when considering Heyting algebras.This fact has also been remarked by Bezhanishvili in [2], and by Božić and Došen in [5].

Preliminaries
Let X,≤ be a poset.The set of all increasing subsets of X is denoted by ᏼ i (X).It is clear that ᏼ i (X) is a bounded distributive lattice under the operations ∪ and ∩.A totally order-disconnected topological space is a triple X,≤,τ such that X,≤ is a poset, X,τ is a topological space, and given x, y ∈ X such that x y there is a clopen increasing set Sergio Arturo Celani 3 U such that x ∈ U and y / ∈ U. A Priestley space is a compact totally order-disconnected topological space.If X is a Priestley space, the set of all clopen increasing sets of X is denoted by D(X).Since D(X) is a ring of sets, then D(X),∪,∩,∅,X is a bounded distributive lattice.It can be proved that D(X) is a subbasis for the topology τ.
If A = A,∨,∧,0,1 is a bounded distributive lattice and X(A) is the set of all prime filters of A, then X(A) is a Priestley space with inclusion as order and with the topology having as subbasis sets of the form σ A (a) = {P ∈ X(A) : a ∈ P} and X(A If X,≤,τ is a Priestley space, then the mapping F X : X → X(D(X)) defined by F X (x) = {U ∈ D(X) : x ∈ U} is an order-isomorphism and a homeomorphism.The set of closed increasing (closed decreasing) subsets of a Priestley space X will be denoted by Ꮿ i (X) (Ꮿ d (X)).The set of open subsets of a Priestley space X will be denoted by ᏻ(X).The set of closed and open (clopen) subsets of X will be denoted by Clop(X).
Let X be a Priestley space and let Y be a subset of X.The closure of Y will be denoted by Cl(Y ).The set of maximal (minimal) elements of Y will be denoted by max Y (minY ).Let us recall that for any nonempty closed subset Y of X, maxY = ∅ (minY = ∅).
Let A be a bounded distributive lattice and let Y be a closed subset of X(A).It is known that is a lattice congruence on A, and the correspondence Y → θ(Y ) establishes an antiisomorphism from the lattice of closed subsets of X(A) onto the lattice of lattice of congruences of A (see [8]).The filter (ideal) generated by a subset H ⊆ A will be denoted by [H)((H]).The lattice of all filters (ideals) of A is denoted by Fi(A)(Id(A)).
Let Y be a subset of a set X.The theoretical complement of Y is denoted by Y c = X − Y .Definition 2.1.An algebra A = A,∨,∧, ,♦,0,1 is a ♦-lattice, if A,∨,∧,0,1 is a bounded distributive lattice and and ♦are unary operations defined on A such that for all a,b ∈ A, A -lattice (♦-lattice) is a bounded distributive lattice A,∨,∧,0,1 with an operator (♦) satisfying the condition (M1) (or (M2)).We will proceed with the duality theory for ♦-lattices.Let R be a binary relation on a set X.For each x ∈ X, let us consider the subset R(x) = {y ∈ X : (x, y) ∈ R}.For each U ⊆ X define the sets (2.2) Definition 2.2.A relational Priestley space [3,11,12,18] is a relational structure X,≤, R ,R ♦ , where X,≤ is a Priestley space, and R and R ♦ are binary relations defined on Let A be a ♦-lattice.We define binary relations R A and R A ♦ on X(A) in the following way: ( , with x, y ∈ X(A).We will also consider the relation For the proof of the following result, see [3,7,12,16,18].Lemma 2.3.Let A be a ♦-lattice.Then for each x ∈ X(A) and for each a ∈ A, (1) a ∈ x if and only if for every y ∈ X(A) such that (x, y) ∈ R A it holds a ∈ y; (2) ♦a ∈ x if and only if there exists y ∈ X(A) such that (x, y) ∈ R A ♦ and a ∈ y.
Theorem 2.4.Let A be a ♦-lattice, then the structure

a relational Priestley space such that the mapping σ
Positive modal algebras are ♦-lattices where the operators and ♦ are connected by two rather weak inequalities (see [6,7,10,11]).Definition 2.5.A ♦-lattice A is a positive modal algebra, or PM-algebra, if the following conditions are satisfied: (1) The dual space of a PM-algebra can be defined as a relational Priestley space X,≤, R ,R ♦ verifying additional conditions.These conditions appear in [11] without detailed proofs.We will provide them for the sake completeness.
Proof.We prove only (1).The proof of ( 2) is similar and left to the reader.

Sergio Arturo Celani 5
Remark 2.7.Let A be a PM-algebra.Let us consider the relation (2.3) So we can deduce that the axioms that relate the operators and ♦ allow us to simplify the dual space of a positive modal algebra by considering the single relation Thus, the associated relational Priestley spaces of PM-algebras (called PM-spaces) (see [7]) can be defined as triples X,≤,R , where X,≤ is a Priestley space and R is a binary relation on X such that (1) for each We note that in a PM-space X,≤,R , for every U ∈ D(X) (for more details on the duality for PM-algebras, see [7]).
Let A be a ♦-lattice.Let a ∈ A. For each n ≥ 0 we define inductively the formula n a as 0 a = a and n+1 a = n a.The formula ♦ n a is defined similarly.We also define the formulas α n (a) and β n (a) as For a binary relation R on a set X, let R n be the binary relation on X defined inductively by R 0 = Id X and R n+1 = R n • R, where Id X is the identity relation on X.The closure reflexive and transitive of R is the relation R * = n≥0 R n .We note that if R is reflexive and transitive, then R * (x) = R(x), for each x ∈ X.

Congruences and subdirectly irreducible algebras
One of the major contributions of the Priestley duality is that it allows us to give an exact characterization of the lattice congruence of a bounded distributive lattice.This characterization has been applied to many classes of algebra, like p-algebras, double palgebras, De Morgan algebras, J-distributive lattices [16], and others.In this section it will be shown that the techniques given in [16] to determine the congruences and the simple and subdirectly irreducible algebras in the variety of J-lattices (or ♦-lattices in our notation) can also be applied to ♦-lattices and PM-algebras.First, we will characterize the closed subsets of a relational Priestley space that correspond to modal congruences, that is, lattice congruences preserving the modal operators.
We will denote by C R♦ (X), C R (X), and C R (X), the lattice of R ♦ -saturated subsets of X, the lattice of R -saturated subsets of X, and the lattice of R-saturated subsets of X, respectively.
We note that the notion of R-saturated is connected with the notion of closed and order-hereditary (or M-hereditary) introduced in [3].
Let A be a ♦-lattice.A lattice congruence A congruence is a lattice congruence θ, that is, a -congruence and a ♦-congruence.We denote by Con(A, )(Con(A,♦)) the lattice of all -congruences (♦-congruences), and by Con(A) the lattice of all congruences.
If A is a lattice, we denote by A d the lattice with the dual order.
The following result is necessary to determine the simple and subdirectly irreducible PM-algebras.
Proof.It is immediate.
(1) Let θ ∈ Con(A, ) and let Y be the closed subset associated with θ.Let x ∈ Y and y ∈ minR A (x). Assume that y / ∈ Y .Since Y is closed, there exist a,b ∈ A such that (a ∧ b,a) ∈ θ(Y ), a ∈ y, and b / ∈ y.We prove that Suppose the contrary.Since y c ∪ {a} is closed under ∨, there exists z ∈ X(A) such that −1 (x) ⊆ z and z ⊆ y.As y is minimal in R A (x), we get z = y.It follows that a ∈ z, which is a contradiction.Thus there exists some q / ∈ y, such that (q ∨ a) ∈ x.Since (a ∧ b,a) ∈ θ(Y ), q ∨ (a ∧ b) , (q ∨ a) ∈ θ. (3. 2) It follows that (q ∨ (a ∧ b)) ∈ x, and this implies that a ∧ b ≤ b ∈ y, which is a contradiction.Thus y ∈ Y and Y ∈ C R (X(A)).Suppose that Y ∈ C R (X(A)).We prove that θ(Y ) preserves the operation .Let a,b ∈ A such that (a,b) ∈ θ(Y ).Suppose that for some x ∈ X(A).Then from Lemma 2.3 there exists y ∈ R A (x) such that b / ∈ y.Since R A (x) is a closed subset of X(A), there exists z ∈ X(A) such that z ∈ minR A (x) and z ⊆ y (for a proof of this fact see [3,Proposition 4.10]).It follows that z Lemma 3.4.Let X,≤,R ,R ♦ be a relational Priestley space.Then the following conditions are mutually exclusive: (1 Proof.Suppose that the conditions (1) and ( 2) are not mutually exclusive.Then there are x, y in X such that We note that Cl R (M(y)) = ∅, because R (y) ∪ R ♦ (y) = ∅.It is clear that x = y, and thus x ∈ Cl R (M(y)).As Cl R (M(y)) is R-saturated we have that M(x) ⊆ Cl R (M(y)), which is a contradiction since by hypothesis Cl R (M(x)) = X and besides Cl R (M(y)) = X.
The proof of the following result is established for ♦-lattices and follows the proof for the characterization of simple and subdirectly irreducible algebras in the variety of ♦-lattices given in [16].It is easy to formulate a similar result for -lattices, and PMalgebras (using Lemma 3.2).Note that the following characterization is more complete than the results obtained in [3], because Birchall only gives a sufficient condition for a ♦-lattice A is subdirectly irreducible.
Theorem 3.5.Let X,≤,R ,R ♦ be the relational Priestley space of a ♦-lattice A. Then (1) A is simple if and only if either for all x ∈ X, R (x

and X is a singleton; (2) A is subdirectly irreducible but nonsimple if and only if one and only one of the following conditions holds true:
Proof.(1) (⇒).Let A be a simple ♦-lattice.Suppose that for all x ∈ X, R (x because A is simple.Now we suppose that x ∈ X and R (x So {x} is an R-saturated subset of X.Since A is simple, we have {x} = X and consequently X is a singleton.
We prove the other implication.Let us assume that R (x for every x ∈ X, and X is a singleton, it is clear that the sets ∅ and X are the only R-saturated subsets, and thus A is simple. (2) From Lemma 3.4 it follows that the conditions (a) and (b) are mutually exclusive.Suppose that A is subdirectly irreducible but not simple.Let Y be the greatest element of C R (X) − {X}.Since A is not simple, Y = ∅.We define the set Since Y is R-saturated and different from X we have Y ⊆ T. Suppose that Y = T. Clearly X − Y is a nonempty open subset of X, and in accordance with the definition of T, we obtain (a).Now, suppose that Y T.
Let us prove the reciprocal.Suppose (a).Let T be the set defined previously.We can see that T is different from X, because otherwise the set {x ∈ X : Cl R (M(x)) = X} would be empty, again by the hypothesis (a).Also it is clear that A is not simple, and T is closed.
We show that T is R-saturated.Let x ∈ T and suppose We see now that T is the greatest element of C R (X Now suppose (b).It is clear that A is not simple.We consider the set

Topological and monadic positive modal algebras
In this section we will introduce the variety of topological positive modal algebras and the variety of monadic positive modal algebras, and we will determine the simple and subdirectly irreducible algebras in these varieties.
Let A be a ♦-lattice.We will write A α ≤ β when the inequality α ≤ β is valid in A. In the next result we will establish that certain additional conditions defined in a ♦- lattice correspond to additional properties defined in the dual space.The proof of these correspondences can be deduced from the results established in [6].Another method where these results can be deduced is taking into account that all these conditions are Sahlqvist equations, and thus the proof also follows by the general results given in [11].
Definition 4.2.Let A be a PM-algebra.Say that A is a topological PM-algebra, or TPMalgebra, if A satisfies the following axioms: (1) a ≤ a, a ≤ ♦a; (2) a ≤ 2 a, ♦ 2 a ≤ ♦a, for all a ∈ A.
Remark 4.3.By Remark 2.7 and Theorem 4.1 we have that the relational Priestley space of a TPM-algebra A is the PM-space X(A),⊆,R A , where R A = R A ∩ R A ♦ is a reflexive and transitive relation.The space associated with a TPM-algebra will be called a TPM-space.Remark 4.4.We note that in a TPM-space X,≤,R , R * (x) = R(x), for each x ∈ X.Moreover, taking into account that min R (x , and that R(x) is a closed subset of X, we have that R(x) is an R-saturated subset, for each x ∈ X.By these facts it is easy to see that the following conditions are equivalent for each x ∈ X, (1) Cl R (M(x)) = X; (2) R(x) = X and Cl R (minX ∪ max X) = X; (3) R (x) = X, and R ♦ (x) = X and Cl R (minX ∪ max X) = X.
We need the following auxiliary result to give the characterization of simple and subdirectly irreducible TPM-algebras.Lemma 4.12.Let A be an MPM-algebra.Then (1) the pair { ,♦} is simple if and only if R A (x) = X(A), for each x ∈ X(A); (2) if A is subdirectly irreducible, then the pair { ,♦} is simple; (3) if A is subdirectly irreducible, and Y is a proper R-saturated subset of X(A), then minX(A) ∪ max X(A) ⊆ Y .
Proof.(1) Assume that the pair { ,♦} is simple.Suppose that there exists , then there exists y ∈ X(A) and there exists b If R A (x) = X(A) for all x ∈ X(A), it is easy to check that the pair { ,♦} is simple.
(2) Assume that A is subdirectly irreducible.We prove that R A (x) = X(A), for each x ∈ X(A).Suppose that there exists x ∈ X(A) such that R A (x) = X(A).Since A is subdirectly irreducible, there exists a greatest proper R-saturated subset Y of X(A).So, there exists y ∈ X(A) − Y .Since R A is an equivalence relation, and the subsets R A (x) and R A (y) are R-saturated subsets of X(A), we get (4.7) Thus, x ∈ R A (y), and this implies that R A (x) = R A (y), which is a contradiction.Therefore, R A (x) = X(A), for each x ∈ X(A).
(3) Let A be subdirectly irreducible, and let Y be a proper R-saturated subset of X(A).Suppose that there exists y ∈ minX(A) such that y / ∈ Y .Then for each x ∈ Y , x y.So for each x ∈ Y there exists a P ∈ A such that a P ∈ x and a P / ∈ y.Then As Y is closed, Y is compact.So there exists a ∈ A such that Y ⊆ σ A (a) and a / ∈ y.Then (a,1) ∈ θ(Y ), and consequently ( a, 1) ∈ θ(Y ).From item (2), the pair { ,♦} is simple.So, ( a, 1) = (0,1) ∈ θ(Y ), that is, Y = ∅, which is a contradiction.Thus, minX(A) ⊆ Y .The proof of the inclusion max X(A) ⊆ Y is similar.Proposition 4.13.Let A be an MPM-algebra.Then, (1) A is simple if and only if Cl(minX(A) ∪ max X(A)) = X(A), and the pair { ,♦} is simple; (2) A is subdirectly irreducible but nonsimple if and only if the pair { ,♦} is simple and there exists x ∈ X(A) with x / ∈ Cl(minX(A)∪max X(A)) such that {x}∪Cl(min X(A) ∪ max X(A)) = X(A).
Sergio Arturo Celani 13 Proof.(1) Since every MPM-algebra is a TPM-algebra, from assertion (1) of Theorem 4.7 and Lemma 4.10, we have the desired result.
(2) that A is subdirectly irreducible but nonsimple.From Lemma 4.12 the pair { ,♦} is simple.From Lemma 4.10, and by part 1, Y = X(A), because A is not simple.
Let Z be the greatest proper R-saturated subset of X(A).Then there exists x ∈ X(A) − Z.We prove that If z = x, then from item (3) of Lemma 4.12 and taking into account that the pair { ,♦} is simple, we get As Z is a greatest proper R-saturated subset of X(A), Y ⊆ Z, and x / ∈ Z, we conclude that Y ∪ {x} = X(A).Conversely, suppose that the pair { ,♦} is simple and there exists − {X(A), ∅}, then Z ⊆ Y .Thus, A is subdirectly irreducible.

Heyting modal algebras
In [14,19] characterizations of subdirectly irreducible algebras for some Heyting algebras with modal operators were given.Similar characterizations for monadic Heyting were given in [2].In this section we will apply the previous results to give a characterization of the simple and subdirectly irreducible for some Heyting algebras with modal operators in terms of relational Priestley spaces.These characterizations are new (as far as we know).We also note that neither [14] nor [19] gives characterization for simple algebras.
An algebra A = A,∨,∧,→,0,1 is a Heyting algebra if A,∨,∧,0,1 is a bounded distributive lattice and → is a binary operation such that for all a,b,c ∈ A, a ∧ b ≤ c iff a ≤ b −→ c.
(5.1) Definition 5.1.A Priestley space X,≤ is a Heyting space, or H-space, if for every U,V ∈ D(X), the set It is clear that if X,≤ is an H-space, then D(X),∪,∩,⇒,∅,X is a Heyting algebra under the implication ⇒.
Proposition 5.2.Let A be Heyting algebra.Then X(A),⊆ is an H-space such that the map σ A : A → D(X(A)) is a Heyting isomorphism, that is, σ A is a lattice isomorphism such that σ A (a) =⇒ σ A (b) = σ A (−→ b). (5.3) Let us recall that under the Priestley duality, the lattice of all filters of a bounded distributive lattice is dually isomorphic to the lattice of all increasing closed subsets of the dual space.Under that isomorphism, any filter F of a bounded distributive lattice A corresponds to the increasing closed and any increasing closed subset Y of X(A) corresponds to the filter (5.5) On the other hand, it is known that there exists a lattice isomorphism between the lattice of all filters and the lattice of all congruences of a Heyting algebra A (see [1]).Under that isomorphism, any congruence θ corresponds to the filter Thus, there exists an isomorphism from the lattice of congruences of a Heyting algebra A onto the lattice of increasing closed subsets of the H-space X(A).
We note that the variety of monadic Heyting algebras is a subvariety of modal Heyting algebras, because a monadic Heyting algebra A [2]  (5.8) We note also that a monadic Heyting algebra is not a monadic positive algebra, since in general the condition (a ∨ b) = a ∨ b is not valid.