Hyperstructure

This article is about a mathematical concept. For the architectural concept, see 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. aha.eled.duth.gr
• Applications of Hyperstructure Theory, Piergiulio Corsini, Violeta Leoreanu, Springer, 2003, ISBN 1-4020-1222-5, ISBN 978-1-4020-1222-8
• Functional Equations on Hypergroups, László, Székelyhidi, World Scientific Publishing, 2012, ISBN 978-981-4407-00-7