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×HP *(H)+ : H \times H \to P^*(H)

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

  1. + is associative (extended to allow addition of subsets of H);
  2. 0+x={x}=x+0,xH;
  3. xH,!yH such that 0x+y (we denote this y as x);
  4. x,y,zH,xy+z implies zxy (where xy 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)