# Hyperstructure

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Algebraic structure equipped with at least one multivalued operation

This article is about a mathematical concept. For the architectural concept, see [Arcology](/source/Arcology).

**Hyperstructures** are [algebraic structures](/source/Algebraic_structure) equipped with at least one [multi-valued](/source/Multi-valued) operation, called a *hyperoperation*. The largest classes of the hyperstructures are the ones called H v {\displaystyle Hv} – structures.

A **hyperoperation** ( ⋆ ) {\displaystyle (\star )} on a [nonempty](/source/Empty_set) [set](/source/Set_(mathematics)) H {\displaystyle H} is a mapping from H × H {\displaystyle H\times H} to the **nonempty [power set](/source/Power_set)** P ∗ ( H ) {\displaystyle P^{*}\!(H)} , meaning the set of all nonempty subsets of H {\displaystyle H} , i.e.

- ⋆ : H × H → P ∗ ( H ) {\displaystyle \star :H\times H\to P^{*}\!(H)}

- ( x , y ) ↦ x ⋆ y ⊆ H . {\displaystyle \quad \ (x,y)\mapsto x\star y\subseteq H.}

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

- A ⋆ B = ⋃ a ∈ A , b ∈ B a ⋆ b {\displaystyle A\star B=\bigcup _{a\in A,\,b\in B}a\star b} and A ⋆ x = A ⋆ { x } , {\displaystyle A\star x=A\star \{x\},\,} x ⋆ B = { x } ⋆ B . {\displaystyle x\star B=\{x\}\star B.}

( H , ⋆ ) {\displaystyle (H,\star )} is a *semihypergroup* if ( ⋆ ) {\displaystyle (\star )} is an [associative](/source/Associative) hyperoperation, i.e. x ⋆ ( y ⋆ z ) = ( x ⋆ y ) ⋆ z {\displaystyle x\star (y\star z)=(x\star y)\star z} for all x , y , z ∈ H . {\displaystyle x,y,z\in H.}

Furthermore, a **hypergroup** is a semihypergroup ( H , ⋆ ) {\displaystyle (H,\star )} , where the [reproduction axiom](https://en.wikipedia.org/w/index.php?title=Reproduction_axiom&action=edit&redlink=1) is valid, i.e. a ⋆ H = H ⋆ a = H {\displaystyle a\star H=H\star a=H} for all a ∈ H . {\displaystyle a\in H.}

## References

- AHA (Algebraic Hyperstructures & Applications). A scientific group at Democritus University of Thrace, School of Education, Greece. [aha.eled.duth.gr](http://aha.eled.duth.gr)

- [Applications of Hyperstructure Theory](https://books.google.com/books?id=uvCrZ3iGur4C), Piergiulio Corsini, Violeta Leoreanu, Springer, 2003, [ISBN](/source/ISBN_(identifier)) [1-4020-1222-5](https://en.wikipedia.org/wiki/Special:BookSources/1-4020-1222-5), [ISBN](/source/ISBN_(identifier)) [978-1-4020-1222-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-1222-8)

- [Functional Equations on Hypergroups](http://www.worldscientific.com/worldscibooks/10.1142/8481), László, Székelyhidi, World Scientific Publishing, 2012, [ISBN](/source/ISBN_(identifier)) [978-981-4407-00-7](https://en.wikipedia.org/wiki/Special:BookSources/978-981-4407-00-7)

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Adapted from the Wikipedia article [Hyperstructure](https://en.wikipedia.org/wiki/Hyperstructure) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Hyperstructure?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.