---
tag: 3C930MVM0T
kind: concept
name: Multialgebra
msc: [08A99, 20N20]
specializes: [QJGPT2W0FT]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperstructure
---

## Summary

A multialgebra is a set with multivalued operations.
Each operation sends a tuple of elements to a nonempty set of elements, not to a single element.
It generalises the [[algebraic structure|QJGPT2W0FT]] of universal algebra, where each value is one element.
Multialgebras include the [[hyperstructures|1HJF40J5AK]] such as hypergroups, and they appear when one passes to quotients, to sets of cosets, or to relations instead of functions.

## Explanations

### L1 human

An ordinary operation, like addition, gives exactly one answer.
A multivalued operation may give several possible answers at once.
For example, on the real numbers let the "combination" of $x$ and $y$ be every number between them.
Then $1$ combined with $3$ is the whole interval $[1, 3]$.
A multialgebra is a set with one or more operations of this kind, of any number of arguments.