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

Abstract

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 Approach to Tense LMn×m-Algebras

“…is a tense LM n×m -space. In addition, from [27] we have that σ A is an LM n×m -isomorphism. Also for all a ∈ A, G R A (σ A (a)) = σ A (G(a)) and H R A−1 (σ A (a)) = σ A (H(a)).…”

“…In [27], A. V. Figallo, I. Pascual and G. Pelaitay determined a topological duality for LM n×m -algebras.…”

“…In this section, we will develop a topological duality for tense n × m-valued Lukasiewicz-Moisil algebras, taking into account the results established by A. V. Figallo, I. Pascual and G. Pelaitay in [27] and the results obtained by A. V. Figallo and G. Pelaitay in [16]. In order to determine this duality, we introduce a topological category whose objects and their corresponding morphisms are described below.…”

“…The main purpose of this paper is to give a topological duality for tense n × m-valued Lukasiewicz-Moisil algebras. In order to achieve this we will extend the topological duality given in [27], for n × m-valued Lukasiewicz-Moisil algebras. In [35] another duality for n×m-valued Lukasiewicz-Moisil algebras was developed, starting from De Morgan spaces and adding a family of continuous functions.…”

“…The paper is organized as follows: In Section 2, we briefly summarize the main definitions and results needed throughout this article. In Section 3, we developed a topological duality for tense n × m-valued Lukasiewicz-Moisil algebras, extending the one obtained in [27] for n × mvalued Lukasiewicz-Moisil algebras. In Section 4, the results of Section 3 are applied.…”