Injective and projective semimodules over involutive semirings

TL;DR

This paper explores the structure of involutive semirings and their semimodules, establishing conditions for subalgebras and demonstrating when injective and projective semimodules coincide, extending MV-algebra theory.

Contribution

It generalizes results on MV-semimodules to involutive semirings, clarifying the role of involution and providing new conditions for semimodule properties.

Findings

Involution is necessary and sufficient for projective and injective semimodules to coincide.

The interval [0,1] can be a subalgebra under specific conditions.

Semiring perspective extends MV-algebra results without the Mundici functor.

Abstract

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called \textit{involutive semirings}. The semiring perspective helps us find a necessary and sufficient condition for the interval $[0,1]$ to be a subalgebra of an involutive residuated lattice. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.