State BL-algebras

The concept of a state MV-algebra was firstly introduced by Flaminio and Montagna in \cite{FlMo0} and \cite{FlMo} as an MV-algebra with internal state as a unary operation. Di Nola and Dvure\v{c}enskij gave a stronger version of a state MV-algebra in \cite{DiDV1}, \cite{DiDV2}. In the present paper we introduce the notion of a state BL-algebra, or more precisely, a BL-algebra with internal state. We present different types of state BL-algebras, like strong state BL-algebras and state-morphism BL-algebras, and we study some classes of state BL-algebras. In addition, we give a sample of important examples of state BL-algebras and present some open problems.