---
tag: 4YSCK1DF26
kind: concept
name: Subvariety
msc: [08B15, 08B99]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
    locator: Subvarieties
---

## Summary

A subvariety of a [[variety|AJNZJD8442]] $\mathcal{V}$ is a variety of the same signature that is contained in $\mathcal{V}$.
It is obtained from $\mathcal{V}$ by adding more identities.
Abelian groups form a subvariety of groups, and distributive lattices form a subvariety of lattices.
The subvarieties of a variety form a lattice, which organises the classification of varieties.

## Explanations

### L1 human

A variety is described by equations; a subvariety obeys those equations and some extra ones.
Adding the equation $xy = yx$ to the group axioms gives the abelian groups.
Adding $x^2 = e$ as well gives the abelian groups in which every element is its own inverse, the vector spaces over the field with two elements.
Each extra equation can only shrink the class.