Hyperstructure

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 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

Applications of Hyperstructure Theory, Piergiulio Corsini, Violeta Leoreanu, Springer, 2003, ISBN 1402012225, 9781402012228