Separable MV-algebras and lattice-groups

TL;DR

This paper characterizes separable MV-algebras as finite products of rational subalgebras of [0,1], linking algebraic structure to geometric frameworks in the context of lattice-ordered groups.

Contribution

It provides a structure theorem for separable MV-algebras, showing they are precisely finite products of rational subalgebras of [0,1], advancing the algebraic geometry of MV-algebras.

Findings

Separable MV-algebras are finite products of rational subalgebras of [0,1]

Establishes a structure theorem for separable MV-algebras

Links algebraic properties to geometric frameworks in MV-algebra theory

Abstract

General theory determines the notion of separable MV-algebra (equivalently, of separable unital lattice-ordered Abelian group). We establish the following structure theorem: An MV-algebra is separable if, and only if, it is a finite product of algebras of rational numbers, i.e., of subalgebras of the MV-algebra $[0,1]\cap\mathbb{Q}$. Beyond its intrinsic algebraic interest, this research is motivated by the long-term programme of developing the algebraic geometry of the opposite of the categroy of MV-algebras, in analogy with the classical case of commutative $K$-algebras over a field $K$.