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. 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.

The term "hyperstructure" can also refer to the densly populated, completely pre-planned city, generally in the form of a single structure or a series of interdependent structures. Paolo Soleri envisions hyperstructures as the archetypal form that would most closely subscribe to his theory of arcology.

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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]