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Hyperstructure

Algebraic structure equipped with at least one multivalued operation

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Source updated April 2, 2026

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 $a\in H.$

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This article is adapted from the Wikipedia article “Hyperstructure”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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