hyperfields

Hyperfields are algebraic structures that generalize fields by allowing addition to be multivalued. Formally, a hyperfield consists of a set H equipped with a hyperaddition ⊕, which maps pairs of elements to nonempty subsets of H, and a single-valued multiplication · that maps H×H to H. There are distinguished elements 0 and 1, with 0 absorbing for multiplication and serving as the neutral element for addition in a hypergroup sense. The nonzero elements of H form a group under multiplication, and the hyperaddition makes (H, ⊕, 0) into a commutative hypergroup. A distributive law a·(b ⊕ c) ⊆ (a·b) ⊕ (a·c) must hold for all a,b,c in H. These axioms generalize the usual field axioms by replacing single-valued addition with a multivalued operation and by preserving a compatible multiplicative structure.

Notable examples illustrate the variety of hyperfields. The Krasner hyperfield is a small, two-element hyperfield that arises in valuation-theoretic contexts and in certain limiting processes. The sign hyperfield encodes only the sign information of elements, with applications in oriented matroids and related combinatorial structures. The tropical hyperfield plays a central role in tropical geometry, encoding valuation data and serving as a bridge between classical algebra and tropical mathematics; it uses a min-plus style addition that is interpreted as a multivalued operation. Other constructions derive hyperfields from valued fields, ordered groups, or equivalence relations that collapse elements by value.

Hyperfields provide a unifying framework for several areas of mathematics, including matroid theory, tropical geometry, and algebraic geometry. They enable a field-like approach to polynomials, roots, and linear spaces in contexts where traditional fields are too rigid, while retaining a rich algebraic structure for systematic study.