Hyperstructure

The hyperstructures are algebraic structures equipped with, at least one multivalued operation, called 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.

References

AHA (Algebraic Hyperstructures & Applications).

A scientific group at Democritus University of Thrace, School of Education, GREECE. aha.eled.duth.gr [1]