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 formula_1 – structures.
A hyperoperation formula_2 on a nonempty set formula_3 is a mapping from formula_4 to the nonempty power set formula_5, meaning the set of all nonempty subsets of formula_3, i.e.
For formula_9 we define
formula_13 is a "semihypergroup" if formula_2 is an associative hyperoperation, i.e. formula_15 for all formula_16 
Furthermore, a hypergroup is a semihypergroup formula_13, where the reproduction axiom is valid, i.e. 
formula_18 for all formula_19