In this paper we propose a new perspective on the theory of MV-algebras based
on the connection between such algebras and idempotent semirings. Such a
viewpoint yields, among other results, interesting representation theorems. We
also present some results of more general interest, such as a matrix-based
characterization of finitely generated projective semimodules over any semiring
and, consequently, the functorial character of the construction of the
Grothendieck group of a semiring.