State BL-algebras

TL;DR

This paper introduces the concept of state BL-algebras, extending the idea of state MV-algebras, and explores various types, examples, and open problems in the field.

Contribution

It defines state BL-algebras, introduces new subclasses like strong state and state-morphism BL-algebras, and provides foundational examples and open questions.

Findings

Different types of state BL-algebras are characterized.

Examples of state BL-algebras are provided.

Open problems in the area are discussed.

Abstract

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.