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 - 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.Comment: This version contains some corrections to some results at the end of Section

Representing quantum structures as near semirings

In this article, we introduce the notion of near semiring with involution. Generalizing the theory of semirings we aim at represent quantum structures, such as basic algebras and orthomodular lattices, in terms of near semirings with involution. In particular, after discussing several properties of near semirings, we introduce the so-called Łukasiewicz near semirings, as a particular case of near semirings, and we show that every basic algebra is representable as (precisely, it is term equivalent to) a near semiring. In the particular case in which a Łukasiewicz near semiring is also a semiring, we obtain as a corollary a representation of MV-algebras as semirings. Analogously, by introducing a particular subclass of Łukasiewicz near semirings, that we termed orthomodular near semirings, we obtain a representation of orthomodular lattices. In the second part of the article, we discuss several universal algebraic properties of Łukasiewicz near semirings and we show that the variety of involutive integral near semirings is a Church variety. This yields a neat equational characterization of central elements of this variety. As a byproduct of such, we obtain several direct decomposition theorems for this class of algebras

The semiring-theoretic approach to MV-algebras: a survey

In this paper we review some of the main achievements of the semiring-theoretic approach to MV-algebras initiated and pursued mainly by the present authors and their collaborators. The survey focuses mainly on the connections between MV-algebras and other theories that such a semiringbased approach enabled, and on an application of such a framework to Digital Image Processing. We also give some suggestions for further developments by stating several open problems and possible research lines.Comment: Published versio