Canonical hypergroup

Definition

A canonical hypergroup is a set $H$ with a hyperoperation $\boxplus$ , an element $0$ , and a unary inverse $a\mapsto-a$ such that, for all $a,b,c\in H$ :

$\boxplus$ is commutative and associative;
$a\boxplus0=\{a\}$ ;
$0\in a\boxplus(-a)$ , and $-a$ is the unique element with this property;
reversibility holds: $c\in a\boxplus b\quad\Longleftrightarrow\quad b\in c\boxplus(-a).$

Reversibility as subtraction

In an abelian group, $c=a+b$ is equivalent to $b=c-a$ . Reversibility is the set-valued version of this implication and prevents a general associative hyperoperation from being called an additive hypergroup without an adequate subtraction law.

Ordinary groups

Every abelian group becomes a canonical hypergroup by replacing each sum $a+b$ by the singleton $\{a+b\}$ . Conversely, a canonical hypergroup all of whose sums are singletons is an abelian group.

Terminology warning

“Hypergroup” also names analytic objects whose products are probability measures. Those are not the Krasner canonical hypergroups used in hyperring theory.

References

Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204. Relevant: Definition 2.1.
Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034. Relevant: multigroups and hyperfields.