Monadic NM-algebras: an algebraic approach to monadic predicate nilpotent minimum logic

Abstract

Abstract In this paper, we further study the variety of monadic nilpotent minimum (NM)-algebras and their corresponding logic. In order to solve the drawback of monadic NM-algebras, we review some well-known classes of monadic t-norm-based fuzzy logical algebras and then revise the axiomatic system of monadic NM-algebras. Then we show that the variety of monadic NM-algebras is the equivalent algebraic semantics of monadic predicate fuzzy logic $\textbf {mNM}_{\forall }$, which is equivalent to the modal fuzzy logic $\textbf {S5(NM)}$. Moreover, we show that the propositional case of the modal fuzzy logic $\textbf {S5(NM)}$, which is $\textbf{S5}^{\prime}\textbf{(NM),}$ is also complete with respect to the variety of monadic NM-algebras in the sense of Blok and Pigozzi and obtain a necessary and sufficient condition for this logic to be semilinear. Finally, we give some representations of monadic NM-algebras. In particular, we give some characterizations of representable and directly indecomposable monadic NM-algebras.