Lattice of subvarieties

Summary

The lattice of subvarieties of a variety $\mathcal{V}$ is the set of all subvarieties of $\mathcal{V}$, ordered by inclusion. It is a complete lattice, and it is dually isomorphic to the lattice of equational theories that contain the theory of $\mathcal{V}$. Its shape records how $\mathcal{V}$ can be cut down by further identities, and its study is a large part of the theory of varieties.