State operators on generalizations of fuzzy structures

摘要

We enlarge the language of Rr-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 state Rr-monoids and state-morphism Rr-monoids. We present basic properties of such algebras. We describe subdirectly irreducible algebras, some generators of the varieties of state-morphism Rr-monoids, and an interplay between states and state operators.