Pseudo MV-algebras and Lexicographic Product

TL;DR

This paper explores algebraic conditions under which pseudo MV-algebras can be represented as intervals in lexicographic products of specific lattice-ordered groups, establishing categorical equivalences.

Contribution

It introduces $(H,u)$-perfect and strong $(H,u)$-perfect pseudo MV-algebras and provides a representation theorem linking them to lexicographic products.

Findings

Characterization of pseudo MV-algebras as lexicographic intervals

Introduction of $(H,u)$-perfect pseudo MV-algebras

Categorical equivalence between strong $(H,u)$-perfect pseudo MV-algebras and $ ext{l}$-groups

Abstract

We study algebraic conditions when a pseudo MV-algebra is an interval in the lexicographic product of an Abelian unital $\ell$-group and an $\ell$-group that is not necessary Abelian. We introduce $(H,u)$-perfect pseudo MV-algebras and strong $(H,u)$-perfect pseudo MV-algebras, the latter ones will have a representation by a lexicographic product. Fixing a unital $\ell$-group $(H,u)$, the category of strong $(H,u)$-perfect pseudo MV-algebras is categorically equivalent to the category of $\ell$-groups.