Non-extendability of Semilattice-valued Measures on Partially Ordered Sets
نویسنده
چکیده
For a poset P and a distributive 〈∨, 0〉-semilattice S, a S-valued poset measure on P is 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 . In relation with congruence lattice representation problems, we consider the problem whether such a measure can be extended to a poset measure μ : P ×P → S, for a larger poset P , such that for all a, b ∈ S and all x ≤ y in P , μ(y, x) = a∨ b implies that there are a positive integer n and a decomposition x = z0 ≤ z1 ≤ · · · ≤ zn = y in P such that either μ(zi+1, zi) ≤ a or μ(zi+1, zi) ≤ b, for all i < n. In this note we prove that this is not possible as a rule, even in case the poset P we start with is a chain and S has size א1. The proof uses a “monotone refinement property” that holds in S provided S is either a lattice, or countable, or strongly distributive, but fails for our counterexample. This strongly contrasts with the analogue problem for distances on (discrete) sets, which is known to have a positive (and even functorial) solution.
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