hyperfield

A hyperfield is an algebraic structure that generalizes a field by allowing the addition operation to be multivalued. It consists of a nonempty set with two operations: a hyperaddition, which assigns to any pair of elements a nonempty subset of the set, and a usual multiplication, which is associative and commutative and has a multiplicative identity. In a hyperfield, every nonzero element has a multiplicative inverse, and the additive structure includes a zero element with 0 + a = {a} for all a. There is a designated additive inverse for each element, denoted −a, such that 0 is contained in a + (−a). The hyperaddition is required to be distributive over multiplication, i.e., a × (b ⊕ c) = (a × b) ⊕ (a × c) for all a, b, c, where ⊕ denotes the hyperaddition.

Examples and varieties of hyperfields illustrate the concept. The Krasner hyperfield, with elements {0, 1}, has 0 + x = x and 1 + 1 = {0, 1}, while 0 × x = 0 and 1 × 1 = 1. The sign hyperfield uses elements {−1, 0, 1} to encode signs of sums, with a hyperaddition reflecting the possible aggregate signs. The tropical hyperfield is built on the real numbers extended by ∞ and uses a multivalued addition that encodes tropical or valuation-theoretic behavior; it underpins tropical geometry and valuated matroids.

Applications and theory: hyperfields provide a unifying language for matroid theory and representability, connecting classical field-based linear algebra with combinatorial structures. A matroid is representable over a hyperfield if it arises from a linear-like structure defined using the hyperfield’s operations; certain matroids correspond to valuated matroids via the tropical hyperfield. Hyperfields also offer frameworks for studying algebraic geometry over nontraditional foundations and exploring generalized notions of field homomorphisms.

Origin and development: the concept was introduced by Marc Krasner in the 1950s as a generalization of fields, with subsequent developments yielding several prominent families and numerous applications.