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 byMcKinsey and Tarski [15] to conduct an algebraic study of topological spaces (see also [4]). In [13],