. R A ] 4 J an 2 00 6 POSET REPRESENTATIONS OF DISTRIBUTIVE SEMILATTICES

We prove that for any 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: (i) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (ii) 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 = x 0 ≤ x 1 ≤ · · · ≤ xn = v such that µ(x i+1 , x i) lies either below a or below b, for all i < n. (iii) The subset {µ(x, 0) | x ∈ P } generates the semilattice S. Furthermore, any finite, bounded subset of P has a join, and P is bounded in case S is bounded. Furthermore, the construction is functorial on lattice-indexed diagrams of finite distributive ∨, 0, 1-semilattices.