Congruence Lattice Problem - Preliminaries

Preliminaries

We denote by Con A the congruence lattice of an algebra A, that is, the lattice of all congruences of A under inclusion.

The following is a universal-algebraic triviality. It says that for a congruence, being finitely generated is a lattice-theoretical property.

Lemma. A congruence of an algebra A is finitely generated if and only if it is a compact element of Con A.

As every congruence of an algebra is the join of the finitely generated congruences below it (e.g., every submodule of a module is the union of all its finitely generated submodules), we obtain the following result, first published by Birkhoff and Frink in 1948.

Theorem (Birkhoff and Frink 1948). The congruence lattice Con A of any algebra A is an algebraic lattice.

While congruences of lattices lose something in comparison to groups, modules, rings (they cannot be identified with subsets of the universe), they also have a property unique among all the other structures encountered yet.

Theorem (Funayama and Nakayama 1942). The congruence lattice of any lattice is distributive.

This says that α ∧ (β ∨ γ) = (α ∧ β) ∨ (α ∧ γ), for any congruences α, β, and γ of a given lattice. The analogue of this result fails, for instance, for modules, as, as a rule, for submodules A, B, C of a given module.

Soon after this result, Dilworth proved the following result. He did not publish the result but it appears as an exercise credited to him in Birkhoff 1948. The first published proof is in Grätzer and Schmidt 1962.

Theorem (Dilworth ≈1940, Grätzer and Schmidt 1962). Every finite distributive lattice is isomorphic to the congruence lattice of some finite lattice.

It is important to observe that the solution lattice found in Grätzer and Schmidt's proof is sectionally complemented, that is, it has a least element (true for any finite lattice) and for all elements a ≤ b there exists an element x with a ∨ x = b and a ∧ x = 0. It is also in that paper that CLP is first stated in published form, although it seems that the earliest attempts at CLP were made by Dilworth himself. Congruence lattices of finite lattices have been given an enormous amount of attention, for which a reference is Grätzer's 2005 monograph.