3 N ov 2 00 7 POSET REPRESENTATIONS OF DISTRIBUTIVE SEMILATTICES
نویسنده
چکیده
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 = x 0 ≤ x 1 ≤ · · · ≤ xn = v such that either µ(x i+1 , x i) ≤ a or µ(x i+1 , x i) ≤ 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 lattice-indexed diagrams of finite distributive ∨, 0, 1-semilattices.
منابع مشابه
N ov 2 00 7 POSET REPRESENTATIONS OF DISTRIBUTIVE
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 pos...
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