Hyperstructure

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

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), where P*(H) denotes 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, hypergroup is a semihypergroup (H, *)where the reproduction axiom is valid, i.e. a*H = H*a = H, for all a of H.

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AHA (Algebraic Hyperstructures & Applications). A scientific group at Democritus University of Thrace, School of Education, GREECE. aha.eled.duth.gr [1]