Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1204.4365

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

A. V. Figallo·I. Pascual y A. Ziliani
View original source ↗Read source document ↗Export metadata ↓

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, $n\geq 2$.

Explore related subjects

math.LO

Keep this discovery

BibTeXRIS

A. V. Figallo, I. Pascual y A. Ziliani. 2012-04-19. Principal and Boolean congruences on θ-valued Lukasiewicz-Moisil algebras. https://arxiv.org/abs/1204.4365

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

There is no maximal $K$-degree

The Kolmogorov complexity of a string characterize how complex it is to describe the string. If every prefix of a real $x$ is more complex to describe than every prefix (of the same length) of real $y$, then it is seen as $x$ is more complex to describe than $y$. It is wondered if there is a real $x$ so that no other reals are strictly more complex (to describe) than $x$. The behavior of Kolmogorov complexity functions generated by reals (namely $n\mapsto$ the minimal description length of the real) is quite chaos. Therefore, it is widely believed that there are many reals that are maximally complex to describe. For instance, it is conjectured that all random enough reals have maximal $K$-degree. In this paper, it is shown that there is no real with maximal $K$-degree. Actually, for almost all real $x$, we can uniformly computably find another real whose $K$-degree is strictly above $x$.

math.LO↗
arXiv2026

Quadruples and cubes

We prove, in $\mathsf{ZFC}$, that the $\lambda$-terraced cube relation fails whenever $\lambda$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $\lambda$. If $\lambda$ is $\aleph_0$ then the pretinent terraced relation has consistency strength of at least one Woodin cardinal. We prove positive polarized relations at a successor and a double successor from wondrous ideals. We show, however, that there are no such ideals over two consecutive cardinals simultaneously.

math.LO↗
arXiv2026

Possibilistic Logic over a Logic of Formal Inconsistency

In this article, we have introduced a new possibilistic logic on a logic of formal inconsistency with the aim of developing a possibility theoretic framework to deal with uncertainty and inconsistency meaningfully without leading to a system collapse. We have discussed the syntax and semantics for this logic and have proved the soundness and completeness theorems. A set of new measures of consistency, contradictoriness, and triviality of a set of formulas have been defined. These have then been put to use in an example to show that this framework can provide better means of machine reasoning.

math.LO↗