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Hyperfields and systems are two algebraic frameworks which have beendeveloped 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 {\ithypersystems}, are characterized by certain elimination axioms. Systems arepreserved under standard algebraic constructions; for instance matrices andpolynomials over hypersystems are systems, but not hypersystems. We illustratethese results by discussing several examples of systems and hyperfields, andconstructions like matroids over systems.

Semiring systems arising from hyperrings