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

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.