---
tag: JXH66D0XD5
kind: concept
name: Minimal variety
msc: [08B15]
specializes: [AJNZJD8442]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Summary

A minimal variety is a nontrivial [[variety|AJNZJD8442]] of algebras whose only proper [[subvariety|4YSCK1DF26]] is the trivial variety.
Minimal varieties are the atoms of the [[lattice of subvarieties|GJAYCVYZYF]] of a variety.
Every nontrivial variety contains at least one minimal variety.

## Explanations

### L1 human

A minimal variety is as small as a variety can be without being trivial.
If one adds to its defining identities any identity that does not already hold in it, only one-element algebras are left.
The variety of Boolean algebras is an example.
Each Boolean algebra with at least two elements contains the two-element Boolean algebra, and that algebra generates the whole variety.