nLab
canonical hypergroup

Idea

A hypergroup is a algebraic structure similar to a group, but where the composition operation does not just take two elements to a single product element in the group, but to a subset of elements of the group.

It is a hypermonoid with additional groupal structure and property.

Definition

A canonical hypergroup is a set, $H$, equipped with a commutative binary operation,

$$+:H\times H\to P^{*}(H)$$+ : H \times H \to P^*(H)

taking values in non-empty subsets of $H$, and a zero element $0\in H$, such that

1. $+$ is associative (extended to allow addition of subsets of $H$);
2. $0+x=\{x\}=x+0,\forall x\in H$;
3. $\forall x\in H,\exists !y\in H$ such that $0\in x+y$ (we denote this $y$ as $-x$);
4. $\forall x,y,z\in H,x\in y+z$ implies $z\in x-y$ (where $x-y$ means $x+(-y)$ as usual).

Examples

The additive structure underlying a hyperring is a canonical hypergroup. See there for more examples.