---
tag: 1HJF40J5AK
kind: concept
name: Hyperstructure
msc: [20N20, 08A99]
specializes: [3C930MVM0T]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperstructure
same_as:
  wikidata: Q5958653
  wikipedia: Hyperstructure
---

## Summary

An algebraic hyperstructure is a set with at least one hyperoperation, a binary operation whose value at two elements is a nonempty set of elements.
The main examples are hypergroups, introduced by Marty in 1934, which generalise [[groups|G5S1M9]] by replacing products with sets of products.
Hyperstructures are special [[multialgebras|3C930MVM0T]], and they arise naturally from quotients of groups by subgroups that are not normal.

## Explanations

### L1 human

In a group, two elements multiply to one element.
In a hypergroup, two elements multiply to a set of elements.
On the real line, let the product of $x$ and $y$ be every number between them; the product of $1$ and $3$ is then $[1, 3]$.
This product is associative in a natural sense.
Also every number $y$ lies in the product of $x$ and $y$, so the products of a fixed $x$ with all numbers cover the line.
These two facts make the line a hypergroup.
Hyperstructures study which facts about groups, rings and fields survive in this setting.