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.Comment: 29 p