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, HH, equipped with a commutative binary operation,

+:H×HP *(H) + : H \times H \to P^*(H)

taking values in non-empty subsets of HH, and a zero element 0H0 \in H, such that

  1. ++ is associative (extended to allow addition of subsets of HH);
  2. 0+x={x}=x+0,xH0 + x = {\{x\}} = x + 0, \forall x \in H;
  3. xH,!yH\forall x \in H, \exists ! y \in H such that 0x+y0 \in x + y (we denote this yy as x-x);
  4. x,y,zH,xy+z\forall x, y, z \in H, x \in y + z implies zxyz \in x - y (where xyx - y means x+(y)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)