Hyperstructure

Definition

Let $H$ be a nonempty set and let $\mathcal{P}^*(H)$ be the set of nonempty subsets of $H$, a part of the power set. A hyperoperation on $H$ is a map $\circ \colon H \times H \to \mathcal{P}^*(H)$ on the Cartesian product. For subsets $X, Y \subseteq H$ write $X \circ Y = \bigcup_{x \in X, y \in Y} x \circ y$, and $x \circ Y = \set{x} \circ Y$. An (algebraic) hyperstructure is a set $H$ with one or more hyperoperations, possibly together with ordinary operations. A set with one hyperoperation $(H, \circ)$ is a hypergroupoid, and it is a hypergroup if:

1. $(x \circ y) \circ z = x \circ (y \circ z)$ for all $x, y, z \in H$ (associativity);
2. $x \circ H = H \circ x = H$ for all $x \in H$ (reproduction axiom).

A hypergroupoid that satisfies clause 1 is a semihypergroup.