Representation of Perfect and Local MV-algebras

TL;DR

This paper presents uniform representation theorems for perfect and local MV-algebras using ultraproducts, and extends these results to local Abelian lattice-ordered groups with strong units, all within ZFC.

Contribution

It provides a comprehensive overview and new uniform representation theorems for local and perfect MV-algebras and related algebraic structures.

Findings

Representation theorems using ultraproducts involving [0,1]

Uniform versions depending on algebra cardinality

Definable constructions within ZFC

Abstract

We describe representation theorems for local and perfect MV-algebras in terms of ultraproducts involving the unit interval [0,1]. Furthermore, we give a representation of local Abelian lattice-ordered groups with strong unit as quasi-constant functions on an ultraproduct of the reals. All the above theorems are proved to have a uniform version, depending only on the cardinality of the algebra to be embedded, as well as a definable construction in ZFC. The paper contains both known and new results and provides a complete overview of representation theorems for such classes.