Abstract

An equivalence between the category of MV-algebras and the category $${{\rm MV^{\bullet}}}$$MV? 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 $${a = \neg \neg a, (a \rightarrow b) \vee (b\rightarrow a) = 1}$$a=¬¬a,(a?b)?(b?a)=1 and $${a \odot (a\rightarrow b) = a \wedge b}$$a?(a?b)=a?b. An object of $${{\rm MV^{\bullet}}}$$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 (A, I), where A is an MV-algebra and I is an ideal of A.

Highlights

  • Cignoli proved in [3] the following facts: (1) K can be extended to a functor from the category of bounded distributive lattices to the category of centered Kleene algebras, (2) there is an equivalence between the category of bounded distributive lattices and the category of centered Kleene algebras whose objects satisfy an additional condition called “interpolation property”, (3) the category of Heyting algebras is equivalent

  • We introduce and study the category IMV: the objects are pairs (A, I), where A ∈ MV and I is an ideal of A, and the morphisms f : (A, I) → (B, J) are morphisms f : A → B in MV which satisfy the condition I ⊆ f −1(J)

  • In 1958 Chang introduced the M V -algebra C [4,5], defined by C = Γ(Z ⊗ Z, (1, 0)), where Z is the set of integer numbers, Z ⊗ Z is the lexicographic product and Γ is the categorical equivalence between –groups with strong unit, and the category MV

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Summary

Introduction

In [1], the previous results were extended giving categorical equivalences for some categories of residuated lattices. An equivalence for the category MV of M V -algebras was developed in [2]. Viglizzo consider the following structure: given an ideal I and a filter F of a bounded distributive lattice A, ∧, ∨, 0, 1, let M (A, I, F ) = {(a, b) ∈ A × A : a ∧ b ∈ I & a ∨ b ∈ F }. We introduce and study the category IMV: the objects are pairs (A, I), where A ∈ MV and I is an ideal of A, and the morphisms f : (A, I) → (B, J) are morphisms f : A → B in MV which satisfy the condition I ⊆ f −1(J). 3 we build up an adjunction between IMV and a new category whose objects are algebras.

Preliminary Definitions and Results
A Connection Between the Categories MV and IMV
The Adjunction
The Categorical Equivalence
Final Remarks
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