Lattice of equational theories

Remarks

The meet of a family of theories is their intersection. The join is the smallest theory containing their union, that is, the set of consequences of the union. So the lattice is a complete lattice, whose top is the theory containing every identity, including $x \approx y$. Identify a theory $T$ with the set of pairs $(s, t)$ such that $s \approx t \in T$. This makes $T$ a fully invariant congruence of the term algebra over $X$, and every fully invariant congruence arises this way. Example: the identities true in all groups form a theory, and so do those true in all abelian groups; the first is contained in the second. Non-example: the set $\set{xy \approx yx}$ alone is not a theory, because it does not contain its consequences $zw \approx wz$ and $x \approx x$. The map $T \mapsto \operatorname{Mod}(T)$ is a dual isomorphism onto the lattice of varieties of signature $\sigma$.