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arXiv · 2207.06739

Semiring systems arising from hyperrings

Marianne Akian·Stephane Gaubert·Louis Rowen
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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.

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Marianne Akian, Stephane Gaubert, Louis Rowen. 2022-07-14. Semiring systems arising from hyperrings. https://arxiv.org/abs/2207.06739

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