On Łukasiewicz-Moisil algebras of fuzzy sets

Abstract

The set (X, J) of fuzzy subsetsf:X→J of a setX can be equipped with a structure of ϑ-valued Łukasiewicz-Moisil algebra, where ϑ is the order type of the totally ordered setJ. Conversely, every Łukasiewicz-Moisil algebra — and in particular every Post algebra — is isomorphic to a subalgebra of an algebra of the form (X, J), whereJ has an order type$\bar \theta$⩾ϑ. The first result of this paper is a characterization of those ϑ-valued Łukasiewicz-Moisil algebras which are isomorphic to an algebra of the form (X, J) (Theorem 1). Then we prove that (X, J) is a Post algebra if and only if the setJ is dually well-ordered (Theorem 2) and we give a characterization of those ϑ-valued Post algebras with are isomorphic to an algebra of the form (X, J) (Theorem 3 and Proposition 2).