Congruence Lattice Problem - A Positive Representation Result For Distributive Semilattices

A Positive Representation Result For Distributive Semilattices

The proof of the negative solution for CLP shows that the problem of representing distributive semilattices by compact congruences of lattices already appears for congruence lattices of semilattices. The question whether the structure of partially ordered set would cause similar problems is answered by the following result.

Theorem (Wehrung 2008). For any distributive (∨,0)-semilattice S, there are a (∧,0)-semilattice P and a map μ : P × P → S such that the following conditions hold:

(1) x ≤ y implies that μ(x,y)=0, for all x, y in P.

(2) μ(x,z) ≤ μ(x,y) ∨ μ(y,z), for all x, y, z in P.

(3) For all x ≥ y in P and all α, β in S such that μ(x,y) ≤ α ∨ β, there are a positive integer n and elements x=z₀ ≥ z₁ ≥ ... ≥ z_2n=y such that μ(z_i,z_i+1) ≤ α (resp., μ(z_i,z_i+1) ≤ β) whenever i < 2n is even (resp., odd).

(4) S is generated, as a join-semilattice, by all the elements of the form μ(x,0), for x in P.

Furthermore, if S has a largest element, then P can be assumed to be a lattice with a largest element.

It is not hard to verify that conditions (1)–(4) above imply the distributivity of S, so the result above gives a characterization of distributivity for (∨,0)-semilattices.