Semiring and semimodule issues in MV-algebras

TL;DR

This paper develops a semiring-theoretic framework for MV-algebras, connecting them with idempotent semirings and introducing MV-semimodules, to deepen the algebraic understanding of MV-algebras.

Contribution

It introduces MV-semimodules and explores their properties, establishing a semiring perspective on MV-algebras and relating them to semilattice endomorphisms and idempotent semifields.

Findings

Representation of MV-algebras as subsemirings of endomorphism semirings

Construction and functoriality of the Grothendieck group for semirings

Analysis of the impact of Mundici's categorical equivalence on MV-semimodules

Abstract

In this paper we propose a semiring-theoretic approach to MV-algebras based on the connection between such algebras and idempotent semirings - such an approach naturally imposing the introduction and study of a suitable corresponding class of semimodules, called MV-semimodules. We present several results addressed toward a semiring theory for MV-algebras. In particular we show a representation of MV-algebras as a subsemiring of the endomorphism semiring of a semilattice, the construction of the Grothendieck group of a semiring and its functorial nature, and the effect of Mundici categorical equivalence between MV-algebras and lattice-ordered Abelian groups with a distinguished strong order unit upon the relationship between MV-semimodules and semimodules over idempotent semifields.