---
tag: 9689SFPVE9
kind: concept
name: Magma
msc: [20N02, 08A62]
specializes: [QJGPT2W0FT]
sources:
  - url: https://en.wikipedia.org/wiki/Magma_(algebra)
same_as:
  wikidata: Q679903
  wikipedia: Magma (algebra)
---

## Summary

A magma is a set with one [[binary operation|0M418FE7BB]] and no further axioms.
It is the most basic [[algebraic structure|QJGPT2W0FT]] with a binary operation: [[semigroups|0XMEM406JM]], [[monoids|0N6R902FG1]], [[groups|G5S1M9]] and quasigroups are magmas with extra conditions.
Older texts call a magma a groupoid, a name that now usually means something else in category theory.

## Explanations

### L1 human

A magma is a set where any two elements can be combined into a third, in a fixed order.
No rule is asked for: the result may depend on how you place brackets and on the order of the two elements.
Subtraction on the integers is an example: $5 - 3$ is an integer, but $(5 - 3) - 1 \ne 5 - (3 - 1)$.
Many familiar structures are magmas with additional rules, so facts about magmas hold for all of them.