Congruence lattice

Remarks

The join $\theta \vee \psi$ is also the transitive closure of $\theta \cup \psi$, that is, the union of the relations $\theta \circ \psi$, $\theta \circ \psi \circ \theta$, and so on. $\operatorname{Con} \mathbf{A}$ is a complete lattice with least element $\Delta_A$ and greatest element $\nabla_A = A \times A$. It is an algebraic lattice, and its compact elements are the finitely generated congruences $\operatorname{Cg}(R)$ with $R$ finite.

Example: for the group $\Z/6\Z$, the congruences correspond to the subgroups $\set{0}$, $\set{0, 3}$, $\set{0, 2, 4}$ and $\Z/6\Z$. So $\operatorname{Con}(\Z/6\Z)$ has four elements, with two incomparable elements in the middle.

Non-example: the union of two congruences need not be their join. On the group $\Z$, the union of congruence modulo $2$ and modulo $3$ contains $(0, 2)$ and $(2, 5)$ but not $(0, 5)$, so it is not transitive.