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,

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

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.

Last revised on July 15, 2010 at 14:30:07.
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