On the geometric theory of local MV-algebras

TL;DR

This paper explores the geometric theory of local MV-algebras, showing that certain quotients are of presheaf type and establishing connections with lattice-ordered abelian groups, with implications for algebraic representations.

Contribution

It demonstrates that quotients of the theory of local MV-algebras are of presheaf type and are Morita-equivalent to expansions of lattice-ordered abelian groups, providing new insights into their structure.

Findings

Quotients of local MV-algebras are of presheaf type.

Morita-equivalence with lattice-ordered abelian groups.

Representation theorem for finitely presentable algebras.

Abstract

We investigate the geometric theory of local MV-algebras and its quotients axiomatizing the local MV-algebras in a given proper variety of MV-algebras. We show that, whilst the theory of local MV-algebras is not of presheaf type, each of these quotients is a theory of presheaf type which is Morita-equivalent to an expansion of the theory of lattice-ordered abelian groups. Di Nola-Lettieri's equivalence is recovered from the Morita-equivalence for the quotient axiomatizing the local MV-algebras in Chang's variety, that is, the perfect MV-algebras. We establish along the way a number of results of independent interest, including a constructive treatment of the radical for MV-algebras in a fixed proper variety of MV-algebras and a representation theorem for the finitely presentable algebras in such a variety as finite products of local MV-algebras.