---
tag: XAC78GN2YJ
kind: concept
name: Commutative magma
msc: [20N02]
specializes: [9689SFPVE9]
sources:
  - url: https://en.wikipedia.org/wiki/Commutative_magma
same_as:
  wikidata: Q5419309
  wikipedia: Commutative magma
---

## Summary

A commutative magma is a [[magma|9689SFPVE9]] whose operation does not depend on the order of the two arguments.
Commutativity does not imply associativity, and commutative magmas are the natural setting to see the two conditions apart.
Examples include the average of two numbers and the operation of "rock, paper, scissors".

## Explanations

### L1 human

In a commutative magma, combining $x$ with $y$ gives the same as combining $y$ with $x$.
But brackets may still matter.
Take the average of two numbers: the average of $0$ and $4$ is the average of $4$ and $0$.
Yet averaging $0$ with $4$ and then with $8$ gives $5$, while averaging $0$ with the average of $4$ and $8$ gives $3$.