State operators on generalizations of fuzzy structures

Abstract

We enlarge the language of R@?-monoids, which are a non-commutative generalizations of both MV algebras and BL algebras, by adding a unary operation that describes algebraic properties of a state (= an analog of probability measures). The resulting algebras are called stateR@?-monoids and state-morphismR@?-monoids. We present basic properties of such algebras. We describe subdirectly irreducible algebras, some generators of the varieties of state-morphism R@?-monoids, and an interplay between states and state operators.