# Hyperstructure

(Redirected from Hypergroup)

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

(*): H × HP*(H): (x, y) → x*yH.

If Α, ΒΗ then we define

A*B = ${\displaystyle \bigcup _{a\in A,b\in B}(a\star b)}$ 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.