On a Definition of a Variety of Monadic ℓ-Groups

Abstract

In this paper we expand previous results obtained in [2] about the study of categorical equivalence between the category IRL 0 of integral residuated lattices with bottom, which generalize MV-algebras and a category whose objects are called c-differential residuated lattices. The equivalence is given by a functor $${{\mathsf{K}^\bullet}}$$ , motivated by an old construction due to J. Kalman, which was studied by Cignoli in [3] in the context of Heyting and Nelson algebras. These results are then specialized to the case of MV-algebras and the corresponding category $${MV^{\bullet}}$$ of monadic MV-algebras induced by "Kalman's functor" $${\mathsf{K}^\bullet}$$ . Moreover, we extend the construction to l-groups introducing the new category of monadic l-groups together with a functor $${\Gamma ^\sharp}$$ , that is "parallel" to the well known functor $${\Gamma}$$ between l and MV-algebras.