What is "hyperstructure"

n. 1 (context mathematics English) Any algebraic structure equipped with a hyperoperation. 2 (cx architecture English) A very densely populated habitat, as in arcology.

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 × H → P*(H): (x, y) → x*y ⊆ H.

If Α, Β ⊆ Η then we define

(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.