Subvariety

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 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.