A Categorical Equivalence Motivated by Kalman’s Construction

Studia Logica 104 (2):185-208 (2016)
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Abstract

An equivalence between the category of MV-algebras and the category MV{{\rm MV^{\bullet}}} MV ∙ is given in Castiglioni et al. :67–92, 2014). An integral residuated lattice with bottom is an MV-algebra if and only if it satisfies the equations a=¬¬a,=1{a = \neg \neg a, \vee = 1} a = ¬ ¬ a, ∨ = 1 and a=ab{a \odot = a \wedge b} a ⊙ = a ∧ b. An object of MV{{\rm MV^{\bullet}}} MV ∙ 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, where A is an MV-algebra and I is an ideal of A.

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