nLab
canonical hypergroup
Contents
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

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

Revised on July 15, 2010 14:30:07
by

Urs Schreiber
(87.212.203.135)