Semirings

TL;DR

This survey explores the development of semiring theory over the past decade, focusing on structures without additive cancellativity and the role of a null ideal in generalizing classical algebraic concepts.

Contribution

It introduces and studies the pair (A, A_0) as a generalization of semirings, extending algebraic theories to broader contexts like polynomials, geometry, and modules.

Findings

Generalization of algebraic structures using null ideals

Extension of polynomial and geometric theories to semirings

Framework for studying semirings via universal algebra

Abstract

We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main feature is a specified ``null ideal'' $\mcA_0$ of a semiring $\mcA,$ taking the place of a zero element, which permits generalizations of the classical algebraic theory to polynomials and their roots, algebraic geometry, matrices, linear algebra, varieties, categories, and module theory. The ``pair'' $(\mcA,\mcA_0)$ is studied along the lines of universal algebra.