State operators on commutative basic algebras

Abstract

Commutative basic algebras which are non-associative generalizations of MV -algebras, are an algebraic counterpart of the non-associative propositional fuzzy logic L CBA . We enlarge the language of commutative basic algebras by adding a unary operation that describes algebraic properties of a state (i.e., an analogue of probability measures). The resulting algebras are state commutative basic algebras which can be taken as an algebraic semantics of a non-associative generalization of Flaminio and Montagna's probabilistic logic. We present basic properties of such algebras and describe an interplay between states and state operators.