Monadic MV-algebras II: Monadic implicational subreducts

Abstract

In this paper, we study the class of all monadic implicational subreducts, that is, the {→, ∀, 1}-subreducts of the class of monadic MV-algebras. We prove that this class is an equational class, which we denote by ML, and we give an equational basis for this variety. An algebra in ML is called a monadic Lukasiewicz implication algebra. We characterize the subdirectly irreducible members of ML and the congruences of every monadic Lukasiewicz implication algebra by monadic filters. We prove that ML is generated by its finite members. Finally, we completely describe the lattice of subvarieties, and we give an equational basis for each proper subvariety.