Monadic BL-algebras: the equivalent algebraic semantics of H\'ajek's monadic fuzzy logic

TL;DR

This paper introduces monadic BL-algebras as an algebraic framework for H"ajek's monadic fuzzy logic, establishing their properties, subvarieties, and their role as semantics for various monadic fuzzy logics.

Contribution

It defines monadic BL-algebras, explores their properties and subvarieties, and connects them to known monadic fuzzy logics like G"odel and ukasiewicz logic.

Findings

Monadic BL-algebras serve as the algebraic semantics for the monadic fragment of Hjek's basic predicate logic.

Subvarieties correspond to semantics of monadic Gdel and ukasiewicz logics.

Complete characterization of totally ordered monadic BL-algebras provided.

Abstract

In this article we introduce the variety of monadic BL-algebras as BL-algebras endowed with two monadic operators $\forall$ and $\exists$. After a study of the basic properties of this variety we show that this class is the equivalent algebraic semantics of the monadic fragment of H\'ajek's basic predicate logic. In addition, we start a systematic study of the main subvarieties of monadic BL-algebras, some of which constitute the algebraic semantics of well-known monadic logics: monadic G\"odel logic and monadic {\L}ukasiewicz logic. In the last section we give a complete characterization of totally ordered monadic BL-algebras.