Semiring and Semimodule Issues in MV-Algebras
Antonio Di Nola, Ciro Russo · Communications in Algebra · 2013
In this article we propose a semiring-theoretic approach to MV-algebras based on the connection between such algebras and idempotent semirings established in [11 Di Nola , A. , Gerla , B. ( 2005 ). Algebras of Łukasiewicz's logic and their semiring reducts. Idempotent Mathematics and Mathematical Physics. 131–144, Contemp. Math., 377, Amer. Math. Soc . [Google Scholar]] and improved in [4 Belluce , L. P. , Di Nola , A. ( 2009 ). Commutative rings whose ideals form an MV-algebra . Mathematical Logic Quarterly. 55 ( 5 ): 468 – 486 .[Crossref], [Web of Science ®] , [Google Scholar]], 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 Mundici , D. ( 1986 ). Interpretation of AF C*-algebras in Łukasiewicz sentential calculus . J. Functional Analysis 65 : 15 – 63 .[Crossref], [Web of Science ®] , [Google Scholar]] upon the relationship between MV-semimodules and semimodules over idempotent semifields.