The congruence lattice of a lattice

The is a temporarty set of notes based on several older (and some ancient) notes. In due course they will be re-done. Much of this is extracted from [5], but I don’t recommend reading that. Let Λ be an arbitrary lattice. We always assume the Λ has a top > and a bottom ⊥. Later we need to assume that Λ is modular, and perhaps even later that it is an idiom, but not just yet. Let Cong(Λ) be the family of lattice congruences of Λ. It is known (and it is almost tivial) that Cong(Λ) is a lattice. We will see that Cong(Λ) is a frame, in fact it is the topology of a space attached to Λ. I am not sure if this fact is well known, even though it is a easy consequence of a very old result of Birkhoff. It is certainly known for a boolean algebra Λ since the frame Cong(Λ) is just OspecΛ, the topology of the spectrum of Λ. More generally, for a distributive lattice Λ the frame Cong(Λ) is just Ospec∗Λ, the patch topology of the spectrum of Λ, that is the topology of the spectrum of the boolean closure of Λ. To analyse Cong(Λ) we view congruences in a different way. Each congruence of Λ is determined by the set of intervals it collapses. Thus Cong(Λ) is equivalent to a lattice C of certain sets C of intervals. To work with this and various associated gadgets we use to larger lattices A,B of sets A ∈ A,B ∈ B of intervals. Later we assume that Λ is complete (and satisfies certain other properties) and then we need a restricted kind of congruence. These give a fourth associated lattice D of sets D ∈ D of intervals. These four kinds of sets of intervals have analogous classes of objects in a module category. Lattice Category