An algebraic study of the logic S5’(BL)

Abstract P. Hájek introduced an S5-like modal fuzzy logic S5(BL) and showed that is equivalent to the monadic basic predicate logic mBL∀. Inspired by the above important results, D. Castaño et al. introduced monadic BL-algebras and their corresponding propositional logic S5’(BL), which is a simplified set of axioms of S5(BL). In this paper, we review the algebraic semantics of S5’(BL) and obtain some new results regarding to monadic BL-algebras. First we recall that S5’(BL) is completeness with respect to the variety 𝕄𝔹𝕃 of monadic BL-algebras and obtain a necessary and sufficient condition for the logic S5’(BL) to be semilinear. Then we study some further algebraic properties of monadic BL-algebras and discuss the relationship between monadic MV-algebras and monadic BL-algebras. Finally we give some characterizations of representable, simple, semisimple and directly indecomposable monadic BL-algebras, which are important members of the variety 𝕄𝔹𝕃. These results also constitute a crucial first step for providing an equivalent algebraic foundation for mBL∀.