Hyperstructure

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 non-empty set $H$ is a mapping from $H\times H$ to power set $P^{*}(H)$ (the set of all non-empty subsets of $H$ ), i.e.

$(\star ):H\times H\to P^{*}(H):(x,y)\mapsto x\star y\subseteq H.$

If $A,B\subseteq H$ then 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. $x\star (y\star z)=(x\star y)\star z,\ \forall x,y,z\in H.$

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

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