This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$-rigid modules for type $A$ and $D$.

The homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsev-like condition, we characterise varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras have TImC. We show that a congruence n-permutable variety has TImC if and only if it is congruence permutable, and construct an idempotent variety with a majority term that fails TImC.

The congruences of a finite sectionally complemented lattice $L$ are not necessarily uniform (any two congruence classes of a congruence are of the same size). To measure how far a congruence $\Theta $ of $L$ is from being uniform, we introduce Spec $\Theta $, the spectrum of $\Theta $, the family of cardinalities of the congruence classes of $\Theta $. A typical result of this paper characterizes the spectrum $S=({{m}_{j}}|j<n)$ of a nontrivial congruence $\Theta $ with the following two properties:

$$({{S}_{1}})\,\,\,\,2\le n\,\,\text{and }n\ne 3.\,\,\,\,$$

$$({{S}_{2}})\,\,\,2\le {{m}_{j}}\,\,\text{and}\,\,{{m}_{j}}\ne 3,\,\,\,\text{for}\,\text{all}\,j<n.$$

We prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.

We prove that any {0,1 }-preserving homomorphism of finite distributive lattices can be realized as the restriction of the congruence relations of a finite planar lattice with no nontrivial automorphisms to an ideal of that lattice, where this ideal also has no nontrivial automorphisms. We also prove that any {0,1 }-preserving homomorphism of finite distributive lattices with more than one element and any homomorphism of groups can be realized, simultaneously, as the restriction of the congruence relations and, respectively, the restriction of the automorphisms of a lattice L to those of an ideal of L; if the groups are both finite, then so is the lattice L.