2. The congruence lattice of an algebra

It is easy to see that an intersection of congruence relations on ${\cal A}$ is again a congruence relation. Therefore the congruence relations on ${\cal A}$ form a complete lattice, Con$({\cal A})$. In fact, Con$({\cal A})$ is simply a sublattice of Equiv$(A)$. Some examples:

(a) For a group $G$, the lattice Con$(G)$ is essentially the same thing as the lattice of normal subgroups, Normal$(G)$.

(b) For a commutative ring $R$, the lattice Con$(R)$ is essentially the same thing as the lattice of ideals of $R$.

(c) The congruence lattice of a four-element chain is the Boolean lattice 2$^ 3$.