On Łukasiewicz-Moisil algebras of fuzzy sets (Q1208118)

The set \((X,J)\) of fuzzy subsets \(f:X\to J\) of a set \(J\) can be equipped with the structure of a \(\theta\)-valued Łukasiewicz-Moisil algebra, where \(\theta\) is the order type of the totally ordered set \(J\). Conversely, every Łukasiewicz-Moisil algebra --- and in particular every Post algebra --- is isomorphic to a subalgebra of an algebra of the form \((X,\overline J)\), where \(\overline J\) has an order type \(\overline\theta\geq\theta\). The first result of this paper is a characterization of those \(\theta\)-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 set \(J\) is dually well-ordered (Theorem 2), and we give a characterization of those \(\theta\)-valued Post algebras which are isomorphic to an algebra of the form \((X,J)\) (Theorem 3 and Proposition 2).