The hyperstructures are algebraic structure

Algebraic structure

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

s 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 (*) on a non-empty set H is a mapping from H x H to power set P*(H) (the set of all non-empty sets of H), i.e.

(*): H x H → P*(H): (x, y) →x*y ⊆ H.

If Α, Β ⊆ Η then we define

A*B =U(a*b) and A*x = A*{x}, x*B = {x}* B .

(Η,*) is a semihypergroup if (*) is an associative hyperoperation i.e. x*( y*z) = (x*y)*z, for all x,y,z of H.
Furthermore, a hypergroup is a semihypergroup (H, *), where the reproduction axiom is valid, i.e. a*H = H*a = H, for all a of H.