# 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,

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

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.

Revised on July 15, 2010 14:30:07 by Urs Schreiber (87.212.203.135)