---
tag: 43M1WFPYGQ
kind: concept
name: Presentation of an algebra
msc: [08B20, 08A50]
sources:
  - url: https://en.wikipedia.org/wiki/Presentation_of_a_group
---

## Summary

A presentation of an algebra in a [[variety|AJNZJD8442]] describes it by generators and relations.
The relations are equations between terms in the generators.
The presented algebra is the quotient of the free algebra on the generators by the smallest congruence that makes the relations true.
Every algebra of the variety has a presentation, and finitely presented algebras are those with finitely many generators and relations.
[[Presentations of groups|0JGEQXQ4AC]] are the case of the variety of groups.

## Explanations

### L1 human

To describe an algebra, name some elements that generate it and list the equations they satisfy.
The algebra is then the "most general" one in which these equations hold, and nothing more.
For example, in commutative rings with identity, one generator $x$ and the relation $x^2 = -1$ present the Gaussian integers $\Z[i]$.
In monoids, generators $a, b$ with relation $ab = ba$ present the monoid $\N^2$ under addition.