Monadic NM-algebras

Abstract: In this paper, we introduce and investigate monadic NM-algebras: a variety of NM-algebras equipped with universal quantifiers. Also, we obtain some conditions under which monadic NM-algebras become monadic Boolean algebras. Besides, we show that the variety of monadic NM-algebras faithfully the axioms on quantifiers in monadic predicate NM logic. Furthermore, we discuss relations between monadic NM-algebras and some related structures, likeness modal NM-algebras and rough approximation spaces. In addition, we investigate monadic filters in monadic NM-algebras. In particular, we characterize simple and subdirectly irreducible monadic NM-algebras and obtain a representation theorem for monadic NM-algebras. Finally, we present monadic NM-logic and prove the (chain) completeness of monadic NM-logic based on monadic NM-algebras. These results constitute a crucial first step for providing a solid algebraic foundation for the monadic predicate NM logic.