The Case for Inverse Semirings

TL;DR

This paper introduces inverse semirings, a new algebraic structure that generalizes semirings with additive monoids as inverse semigroups, and explores their properties, examples, and connections to other algebraic theories.

Contribution

It defines inverse semirings, studies their fundamental properties, and connects them to existing algebraic structures like rings and inverse semigroups, including applications to bounded polynomials.

Findings

Inverse semirings include important classes of semirings and new examples.

Downward-closed submodules are exactly kernels in inverse semirings.

Connections established between inverse semirings and E-unitary inverse semigroups.

Abstract

A semiring generalises the notion of a ring, replacing the additive abelian group structure with that of a commutative monoid. In this paper, we study a notion positioned between a ring and a semiring -- a semiring whose additive monoid is a commutative inverse semigroup. These inverse semirings include some important classes of semirings, as well as some new motivating examples. We devote particular attention to the inverse semiring of bounded polynomials and argue for their computational significance. We then prove a number of fundamental results about inverse semirings, their modules and their ideals. Parts of the theory show strong similarities with rings, while other parts are akin to the theory of idempotent semirings or distributive lattices. We note in particular that downward-closed submodules are precisely kernels. We end by exploring a connection to the theory of E-unitary inverse semigroups.