Semiring and semimodule issues in MV-algebras

In this paper we propose a semiring-theoretic approach to MV-algebras based on the connection between such algebras and idempotent semirings established in [11] and improved in [4] — such an approach naturally imposing the introduction and study of a suitable corresponding class of semimodules, called MV-semimodules. Besides some basic yet fundamental results of more general interest for semiring theory we present several results addressed toward a semiring theory for MV-algebras. In particular we give a representation of MV-algebras as a subsemiring of the endomorphism semiring of a semilattice, show how to construct the Grothendieck group of a semiring and prove that this construction has a functorial nature. We also study the effect of Mundici categorical equivalence between MV-algebras and lattice-ordered Abelian groups with a distinguished strong order unit [31] upon the relationship between MV-semimodules and semimodules over idempotent semifields.