Hyperoperation - Knowlpedia

Definition

A binary hyperoperation on a set $X$ is a map

\boxplus:X\times X\longrightarrow \mathcal P^*(X),

where $\mathcal P^*(X)$ is the collection of nonempty subsets of $X$ . Thus $x\boxplus y$ is a nonempty set of possible sums, not one chosen element.

Extension to subsets

For nonempty subsets $A,B\subseteq X$ , extend the operation by

A\boxplus B=\bigcup_{a\in A,\,b\in B}(a\boxplus b).

Expressions such as $(a\boxplus b)\boxplus c$ use this extension. Associativity means equality of subsets $(a\boxplus b)\boxplus c=a\boxplus(b\boxplus c)$ , not merely that the two subsets intersect.

Ordinary operations as a special case

An ordinary binary operation $*$ determines the singleton-valued hyperoperation $a\boxplus b=\{a*b\}$ . Hyperoperations therefore include ordinary operations, but a genuinely multivalued operation cannot be treated as a function $X\times X\to X$ without discarding information.

References

Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204. Relevant: §2, hyperoperations and hypergroups.
Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034. Relevant: §3, multivalued addition.