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 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 × HP*(H): (x, y) → x*yH.

If Α, ΒΗ 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[edit]

  • AHA (Algebraic Hyperstructures & Applications). A scientific group at Democritus University of Thrace, School of Education, Greece. aha.eled.duth.gr