Hyperstructure

Hyperstructures are algebraic structures equipped with at least one multi-valued operation, called a hyperoperation. The largest classes of the hyperstructures are the ones called $$Hv$$ – structures.

A hyperoperation $$(\star)$$ on a nonempty set $$H$$ is a mapping from $$H \times H$$ to the nonempty power set $$P^{*}\!(H)$$, meaning the set of all nonempty subsets of $$H$$, i.e.


 * $$\star: H \times H \to P^{*}\!(H)$$
 * $$\quad\ (x,y) \mapsto x \star y \subseteq H.$$

For $$A,B \subseteq H$$ we define


 * $$ A \star B = \bigcup_{a \in A,\, b \in B} a \star b$$ and $$ A \star x = A \star \{ x \},\,$$ $$x \star B = \{x\} \star B.$$

$$ (H, \star ) $$ is a semihypergroup if $$(\star)$$ is an associative hyperoperation, i.e. $$ x \star (y \star z) = (x \star y)\star z$$ for all $$x, y, z \in H.$$

Furthermore, a hypergroup is a semihypergroup $$ (H, \star ) $$, where the reproduction axiom is valid, i.e. $$ a \star H = H \star a = H$$ for all $$a \in H.$$