Lattice of subvarieties

Remarks

An intersection of varieties is a variety, since it is defined by the union of their identities. So $L(\mathcal{V})$ is a complete lattice, with least element the trivial variety and greatest element $\mathcal{V}$. The maps $\mathcal{W} \mapsto \operatorname{Id}(\mathcal{W})$ and $T \mapsto \operatorname{Mod}(T)$ are inverse dual isomorphisms between $L(\mathcal{V})$ and the lattice of equational theories that contain $\operatorname{Id}(\mathcal{V})$. Example: for the variety of Boolean algebras, $L(\mathcal{V})$ has exactly two elements, the trivial variety and $\mathcal{V}$ itself. Example: for abelian groups, the subvarieties are $\mathcal{A}_n = \operatorname{Mod}(nx \approx 0)$ for $n \ge 1$ and the whole variety, and $\mathcal{A}_n \subseteq \mathcal{A}_m$ if and only if $n \mid m$. Non-example: the join is not the union in general, since the union of $\mathcal{A}_2$ and $\mathcal{A}_3$ does not contain $\Z / 2\Z \times \Z / 3\Z \in \mathcal{A}_6$.