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.

Definitions: Hyperoperation (∗) on a non-empty set Η is a mapping from Η x Η to set ℘*(Η), where ℘*(Η) denotes the set of all non-empty sets of H, i.e. ∗: H x H → ℘*(Η): (x, y) ↦x ∗ y ⊆ H. If Α, Β ⊆ Η then we define Α∗Β = (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, ∀ x,y,z Ξ H. Furthermore, hypergroup is a semihypergroup (Η, ∗) where the reproduction axiom is valid, i.e. a∗H = H∗a = H, ∀a Ξ H.

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AHA (Algebraic Hyperstructures & Applications). A scintific group at Democritus University of Thrace, School of Education. [aha.eled.duth.gr]