Hyperstructure

{{Short description|Algebraic structure equipped with at least one multivalued operation}} {{about|a mathematical concept|the architectural concept|Arcology}} '''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== {{reflist}} *AHA (Algebraic Hyperstructures & Applications). A scientific group at Democritus University of Thrace, School of Education, Greece. [http://aha.eled.duth.gr aha.eled.duth.gr] *[https://books.google.com/books?id=uvCrZ3iGur4C Applications of Hyperstructure Theory], Piergiulio Corsini, Violeta Leoreanu, Springer, 2003, {{ISBN|1-4020-1222-5}}, {{ISBN|978-1-4020-1222-8}} *[http://www.worldscientific.com/worldscibooks/10.1142/8481 Functional Equations on Hypergroups], László, Székelyhidi, World Scientific Publishing, 2012, {{ISBN|978-981-4407-00-7}}

Category:Abstract algebra