Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective

TL;DR

This paper explores the deep categorical relationships between lattice-ordered abelian groups and perfect MV-algebras using topos theory, revealing Morita-equivalence and various levels of bi-interpretability.

Contribution

It generalizes existing categorical equivalences, identifies levels of bi-interpretability, and provides new insights into the syntax and semantics of these algebraic theories via topos-theoretic invariants.

Findings

Establishes Morita-equivalence between the theories

Identifies three levels of bi-interpretability for specific formulas

Provides a concrete representation of finitely presentable models

Abstract

We establish, generalizing Di Nola and Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-intepretability holding for particular classes of formulas: irreducible formulas, geometric sentences and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.