Grätzer–Schmidt theorem

Remarks

A congruence of an algebra $A$ is an equivalence relation on $A$ that is compatible with every operation of $A$. For every algebra $A$, the lattice $\operatorname{Con} A$ is algebraic: its compact elements are the finitely generated congruences, and every congruence is the join of the congruences generated by single pairs. So the theorem says that the algebraic lattices are exactly the congruence lattices of algebras, up to isomorphism. The proof is an explicit construction: the operations of $A$ are chosen so that the finitely generated congruences of $A$ behave like the compact elements of $L$. The algebra produced may have infinitely many operations; the representation of finite lattices as congruence lattices of finite algebras is a different and harder question.

Going deeper

For a student

In a course on universal algebra this theorem closes the description of congruence lattices. It sits next to the Birkhoff–Frink theorem, which says the same thing for lattices of subalgebras. For special classes of algebras the question changes: the congruence lattice of a lattice is always distributive, and which distributive algebraic lattices occur there is the congruence lattice problem.