Valued Lukasiewicz-Moisil algebras with and without negation

Abstract

Philosophical problems arising from the idea that there are statements which are neither true nor false, led to the formulation of many–valued logics by Lukasiewicz. Since then, plenty of research has been developed in this area. In 1968, G. Moisil found an example which gave him the motivation he had been looking for in order to legitimate the introduction and study of infinitely–valued Lukasiewicz algebras, so he defined θ − valued Lukasiewicz algebras, where θ is the order type of a chain. In this article, we determine a topological duality for θ − valued Lukasiewicz algebras (or Lk θ − algebras) equivalent to the one given by Filipoiu in 1980. Not only does the duality enable us to obtain a description of the Lk θ − congruences on an Lk θ − algebra, but also to characterize the subdirectly irreducible Lk θ − algebras. Furthermore, we extend the above study to the case of Lk θ − algebras with negation arriving through a different method at the results indicated by V. Boisescu et al ( Lukasiewicz–Moisil Algebras, Annals of Discrete Mathematics 49, North–Holland, 1991).