Hyperstructure Explained

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

(\star) on a nonempty set H is a mapping from H x H to the nonempty power set P ^{*} (H) , meaning the set of all nonempty subsets of H , i.e.\star: H x H \to P ^{*} (H) (x,y) \mapsto x \star y \subseteq H.

For

A,B \subseteq H we defineA \star B = c up _{a} a \star b and A \star x = A \star \{ x \}, x \star B = \{x\} \star B. (H, \star ) is a semihypergroup if (\star) is an associative hyperoperation, i.e. x \star (y \star z) = (x \star y)\star z for all x, y, z \in H.

Furthermore, a hypergroup is a semihypergroup

(H, \star ) , where the reproduction axiom is valid, i.e. a \star H = H \star a = H for all a \in 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,,
• Functional Equations on Hypergroups, László, Székelyhidi, World Scientific Publishing, 2012,