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 <math>Hv</math> – structures.

A hyperoperation <math>(\star)</math> on a nonempty set <math>H</math> is a mapping from <math>H \times H</math> to the nonempty power set <math>P^{*}\!(H)</math>, meaning the set of all nonempty subsets of <math>H</math>, i.e.

:<math>\star: H \times H \to P^{*}\!(H)</math>

:<math>\quad\ (x,y) \mapsto x \star y \subseteq H.</math>

For <math>A,B \subseteq H</math> we define

:<math> A \star B = \bigcup_{a \in A,\, b \in B} a \star b</math> and <math> A \star x = A \star \{ x \},\,</math> <math>x \star B = \{x\} \star B.</math>

<math> (H, \star ) </math> is a semihypergroup if <math>(\star)</math> is an associative hyperoperation, i.e. <math> x \star (y \star z) = (x \star y)\star z</math> for all <math>x, y, z \in H.</math>

Furthermore, a hypergroup is a semihypergroup <math> (H, \star ) </math>, where the reproduction axiom is valid, i.e.

<math> a \star H = H \star a = H</math> for all <math>a \in H.</math>

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,