Poset representations of distributive semilattices

We prove that for every distributive 〈∨, 0〉-semilattice S, there are a meet-semilattice P with zero and a map μ : P × P → S such that μ(x, z) ≤ μ(x, y)∨μ(y, z) and x ≤ y implies that μ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (P1) μ(v, u) = 0 implies that u = v, for all u ≤ v in P . (P2) For all u ≤ v in P and all a,b ∈ S, if μ(v, u) ≤ a ∨ b, then there are a positive integer n and a decomposition u = x0 ≤ x1 ≤ · · · ≤ xn = v such that either μ(xi+1, xi) ≤ a or μ(xi+1, xi) ≤ b, for each i < n. (P3) The subset {μ(x, 0) | x ∈ P} generates the semilattice S. Furthermore, every finite, bounded subset of P has a join, and P is bounded in case S is bounded. Furthermore, the construction is functorial on latticeindexed diagrams of finite distributive 〈∨, 0, 1〉-semilattices.