Principal and Boolean congruences on θ-valued Lukasiewicz-Moisil algebras

Abstract

The first system of many-valued logic was introduced by J. Lukasiewicz, his motivation was of philosophical nature as he was looking for an interpretation of the concepts of possibility and necessity. Since then, plenty of research has been developed in this area. In 1968, when Gr.C. Moisil came across Zadeh's fuzzy set theory he found 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 (or LMθ-algebras, for short) (without negation), where θis the order type of a chain. In this article, our main interest is to investigate the principal and Boolean congruences and θ-congruences on LMθ-algebras. In order to do this we take into account a topological duality for these algebras obtained in (A.V. Figallo, I. Pascual, A. Ziliani, A Duality for θ-Valued Lukasiewicz-Moisil Algebras and Aplicattions. J. LMult- Valued Logic & Soft Computing. Vol. 16, pp 303-322. (2010). Furthermore, we prove that the intersection of two principal ?-congruences is a principal one. On the other hand, we show that Boolean congruences are both principal congruences and θ-congruences. This allowed us to establish necessary and sufficient conditions so that a principal congruence is a Boolean one. Finally, bearing in mind the above results, we characterize the principal and Boolean congruences on n-valued Lukasiewicz algebras without negation when we consider them as LMθ-algebras in the case that θis an integer n, $n\ge 2$.