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Journal Articles Journal of Pure and Applied Algebra Year : 2024

Semiring systems arising from hyperrings

Marianne Akian
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Stephane Gaubert


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. We show that, 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.

Dates and versions

hal-03792658 , version 1 (30-09-2022)




Marianne Akian, Stephane Gaubert, Louis Rowen. Semiring systems arising from hyperrings. Journal of Pure and Applied Algebra, 2024, 228 (6), pp.107584. ⟨10.1016/j.jpaa.2023.107584⟩. ⟨hal-03792658⟩
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