Semiring systems arising from hyperrings

TL;DR

This paper explores the relationship between hyperrings, hyperfields, and systems, characterizing hypersystems with elimination axioms and examining their algebraic properties and applications.

Contribution

It introduces hypersystems as a class of systems representing hyperrings and hyperfields, with a focus on their axiomatic characterization and algebraic behavior.

Findings

Hypersystems are characterized by specific elimination axioms.

Systems are preserved under algebraic constructions like matrices and polynomials.

Examples include matroids over systems and various hyperfield structures.

Abstract

Hyperfields and systems are two algebraic frameworks which have been developed to provide a unified approach to classical and tropical structures. All hyperfields, and more generally hyperrings, can be represented by systems. Conversely, we show that the systems arising in this way, called {\it hypersystems}, are characterized by certain elimination axioms. Systems are preserved under standard algebraic constructions; for instance matrices and polynomials over hypersystems are systems, but not hypersystems. We illustrate these results by discussing several examples of systems and hyperfields, and constructions like matroids over systems.