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 $Hv$ – structures.

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

: $\star: H \times H \to P^{*}\!(H)$

: $\quad\ (x,y) \mapsto x \star y \subseteq H.$

For $A,B \subseteq H$ we define

: $A \star B = \bigcup_{a \in A,\, b \in B} a \star b$ and $A \star x = A \star \{ x \},\,$ $x \star B = \{x\} \star B.$

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

Furthermore, a hypergroup is a semihypergroup $(H, \star )$ , where the reproduction axiom is valid, i.e.

$a \star H = H \star a = H$ for all $a \in H.$

References

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