Łukasiewicz-Moisil Relation Algebras

Abstract

We introduce Lukasiewicz-Moisil relation algebras, obtained by considering a relational dimension over Lukasiewicz-Moisil algebras. We prove some arithmetical properties, provide a characterization in terms of complex algebras, study the connection with relational Post algebras and characterize the simple structures and the matrix relation algebras. A. Popescu[11] and to the MV approach from [32] and [33]. Unlike in these latter cases, the connection with classical logic, being explicitly stated by means of the Chrysippian nuances, allows one to point out certain interesting similarities between LM and classical relational structures. Therefore, we consider that an abstract study of LM-based relation algebras is useful, not only for enriching the many-valued relational algebraic theory developed in [11,32,33], but also for raising the issue of communication with classical relational structures. In this paper, we intend to study these structures, using the following route.Section 2 is a preliminary section on Lukasiewicz-Moisil algebras and classical relation algebras. In Section 3, we introduce LM-relation algebras (LMRAs) and prove some arithmetical properties of these structures. Section 4 studies LM-based complex algebras. There are proved a characterization of complex algebras, as well as a representation theorem similar to the one from the classical case. In Section 5, using the adjunction to Boolean relation algebras preserved from the "non-relational" case, we prove some structural properties of LMRAs -among them, the fact that "simple" is equivalent to "subdirectly irreducible" and a representation theorem for concrete LMRAs, that is, LMRAs isomorphic to concrete structures of many-valued relations on some set X. In Section 6, we study matrix relation algebras and prove that they form an elementary class. A brief discussion of related work ends the paper.