---
tag: KNDSJAYMSY
kind: concept
name: Unital magma
msc: [20N02]
specializes: [9689SFPVE9]
sources:
  - url: https://en.wikipedia.org/wiki/Magma_(algebra)
---

## Summary

A unital magma is a [[magma|9689SFPVE9]] with an [[identity element|07WYQPB8E3]], an element that leaves every element unchanged when combined with it on either side.
The identity of a unital magma is unique.
A [[monoid|0N6R902FG1]] is exactly an associative unital magma, and loops in the theory of quasigroups are unital magmas with unique solutions of equations.

## Explanations

### L1 human

A unital magma has a neutral element $e$: combining $e$ with any $x$, in either order, gives back $x$.
No other rule is asked for.
For example, the integers with $x * y = x + y + xy$ have neutral element $0$, since $0 * y = y$ and $x * 0 = x$.
If two elements $e$ and $e'$ were both neutral, then $e = e * e' = e'$, so there is only one.