A Categorical Equivalence Motivated by Kalman’s Construction

Abstract

An equivalence between the category of MV-algebras and the category ${{\rm MV^{\bullet}}}$ is given in Castiglioni et al. (Studia Logica 102(1):67–92, 2014). An integral residuated lattice with bottom is an MV-algebra if and only if it satisfies the equations ${a = \neg \neg a, (a \rightarrow b) \vee (b\rightarrow a) = 1}$ and ${a \odot (a\rightarrow b) = a \wedge b}$. An object of ${{\rm MV^{\bullet}}}$ is a residuated lattice which in particular satisfies some equations which correspond to the previous equations. In this paper we extend the equivalence to the category whose objects are pairs (A, I), where A is an MV-algebra and I is an ideal of A.