A new topological duality for n × m-valued Łukasiewicz–Moisil algebras

In 2000, Figallo and Sanza introduced [Formula: see text]-valued Łukasiewicz–Moisil algebras which are both a particular case of matrix Łukasiewicz algebras [W. Suchoń, Matrix Łukasiewicz Algebras, Rep. Math. Logic 4 (1975) 91–104] and a generalization of [Formula: see text]-valued Łukasiewicz–Moisil algebras. It is worth noting that unlike what happens in [Formula: see text]-valued Łukasiewicz–Moisil algebras, generally the De Morgan reducts of [Formula: see text]-valued Łukasiewicz–Moisil algebras are not Kleene algebras. Furthermore, in [C. Sanza, [Formula: see text]-valued Łukasiewicz algebras with negation, Rep. Math. Logic 40 (2006) 83–106] an important example which legitimated the study of this class of algebras is provided. In this paper, we continue the study of [Formula: see text]-valued Łukasiewicz–Moisil algebras (or [Formula: see text]-algebras). More precisely, we determine a new topological duality for these algebras. By means of this duality we characterize the congruences and specially the maximal congruences on [Formula: see text]-algebras. Then these characterizations allow us to assert that [Formula: see text]-algebras are semisimples and obtain a new description of subdirectly irreducible [Formula: see text]-algebras.

A topological duality for strong Boolean posets

In this paper, we define a new class of posets which are complemented and ideal-distributive, we call these posets strong Boolean. This definition is a generalization of Boolean lattices on posets, and is different from Boolean posets. We give a topology on the set of all prime Frink ideals in order to obtain the Stone’s topological representation for strong Boolean posets. A discussion of a duality between the categories of strong Boolean posets and BP-spaces is also presented.