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 ${\displaystyle Hv}$ – structures.

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

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

${\displaystyle \quad \ (x,y)\mapsto x\star y\subseteq H.}$

For ${\displaystyle A,B\subseteq H}$ we define

${\displaystyle A\star B=\bigcup _{a\in A,\,b\in B}a\star b}$ and ${\displaystyle A\star x=A\star \{x\},\,}$ ${\displaystyle x\star B=\{x\}\star B.}$

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

Furthermore, a hypergroup is a semihypergroup ${\displaystyle (H,\star )}$, where the reproduction axiom is valid, i.e. ${\displaystyle a\star H=H\star a=H}$ for all ${\displaystyle a\in H.}$