Hyperstructure

This article is about a mathematical concept. For the architectural concept, see 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

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

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 1-4020-1222-5, ISBN 978-1-4020-1222-8

• Functional Equations on Hypergroups, László, Székelyhidi, World Scientific Publishing, 2012, ISBN 978-981-4407-00-7