Semiring systems arising from hyperrings

Marianne Akian, Stephane Gaubert, Louis Rowen

Research output: Contribution to journalArticlepeer-review

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 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.

Original languageEnglish
Article number107584
JournalJournal of Pure and Applied Algebra
Volume228
Issue number6
DOIs
StatePublished - Jun 2024

Bibliographical note

Publisher Copyright:
© 2023 Elsevier B.V.

Funding

The research of the third author was supported by the ISF grant 1994/20 and the Anshel Peffer Chair.

FundersFunder number
Israel Science Foundation1994/20

    Keywords

    • Fuzzy ring
    • Hyperfield
    • Metatangible
    • Semiring
    • System
    • Tropical algebra

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