Hyperrings

Hyperrings are a generalization of rings in which the addition operation is a hyperoperation rather than a single-valued operation. In a hyperring, the set R is equipped with a multivalued addition, denoted by a ⊕ b, which returns a nonempty subset of R, and a ordinary multiplication, denoted by ·, which is a binary operation. The structure is intended to mirror many ring-like properties while allowing the sum of two elements to have multiple possible outcomes.

Formally, a hyperring (R, ⊕, ·, 0, 1) consists of a nonempty set R with a hyperoperation ⊕: R × R → P(R) and a multiplication ·: R × R → R, along with distinguished elements 0 and 1, satisfying: (R, ⊕) is a canonical hypergroup with neutral element 0 and additive inverses such that for each a there exists −a with 0 ∈ a ⊕ (−a); the hyperoperation is associative in the hypergroup sense; (R, ·) is a monoid with identity 1; and distributivity holds in the sense that a · (b ⊕ c) = (a · b) ⊕ (a · c) and (b ⊕ c) · a = (b · a) ⊕ (c · a) for all a,b,c ∈ R. It is also common to require 0 · a = 0 for all a.

Variants include commutative hyperrings (where multiplication is commutative) and hyperfields, which are hyperrings with multiplicative inverses for all nonzero elements. The simplest example is the Krasner hyperfield, with elements {0, 1} and addition defined so that 1 ⊕ 1 = {0, 1}. Hyperrings have applications in tropical and algebraic geometry, valuation theory, and the study of algebraic structures over hyperfields. Hyperstructures were introduced by F. Marty in the 1930s and have since been developed in various directions.