Simple and subdirectly irreducibles bounded distributive lattices with unary operators

Sergio Arturo Celani

doi:10.1155/ijmms/2006/21835

Abstract

We characterize the simple and subdirectly irreducible distributive algebras in some varieties of distributive lattices with unary operators, including topological and monadic positive modal algebras. Finally, for some varieties of Heyting algebras with operators we apply these results to determine the simple and subdirectly irreducible algebras.

Highlights

Distributive lattices with operators (DLO) are a natural generalization of the notion of Boolean algebras with operators
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
The algebraic semantic of this fragment is the variety of positive modal algebras introduced in [10], and further studied by means of topological methods in [7], and in [6] by methods from abstract algebraic logic

Summary

Introduction

Distributive lattices with operators (DLO) are a natural generalization of the notion of Boolean algebras with operators. 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. 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.

Simple and subdirectly irreducibles bounded

Preliminaries

Congruences and subdirectly irreducible algebras

Topological and monadic positive modal algebras

Heyting modal algebras