---
tag: J8FCCSQ2G6
kind: definition
name: Subvariety
msc: [08B15, 08B99]
defines: 4YSCK1DF26
uses: [AJNZJD8442, PKD440G2XW]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
    locator: Subvarieties
---

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras of signature $\sigma$.
A *subvariety* of $\mathcal{V}$ is a variety $\mathcal{W}$ of signature $\sigma$ such that $\mathcal{W} \subseteq \mathcal{V}$.

## Remarks

If $\mathcal{V} = \operatorname{Mod}(\Sigma)$, the subvarieties of $\mathcal{V}$ are exactly the classes $\operatorname{Mod}(\Sigma \cup \Gamma)$ for sets $\Gamma$ of [[identities|PKD440G2XW]] of signature $\sigma$.
Example: abelian groups form a subvariety of the variety of groups, since they are the groups that satisfy $xy \approx yx$.
Non-example: the class of finite groups is contained in the variety of groups but is not a subvariety.
It is not closed under products: $\Z$ is a subgroup of $\prod_{n \ge 1} \Z / n\Z$, generated by the element $(1, 1, \ldots)$.
Every variety has itself and the trivial variety, of one-element algebras, as subvarieties.