# Non-extendability of semilattice-valued measures on partially ordered
sets

Research paper by **Friedrich Wehrung**

Indexed on: **28 Feb '06**Published on: **28 Feb '06**Published in: **Mathematics - General Mathematics**

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#### Abstract

For a distributive join-semilattice S with zero, a S-valued poset measure on
a poset P is a map m:PxP->S such that m(x,z) <= m(x,y)vm(y,z), and x <= y
implies that m(x,y)=0, for all x,y,z in P. In relation with congruence lattice
representation problems, we consider the problem whether such a measure can be
extended to a poset measure m*:P*xP*->S, for a larger poset P*, such that for
all a,b in S and all x <= y in P*, m*(y,x)=avb implies that there are a
positive integer n and a decomposition x=z\_0 <= z\_1 <= ... <= z\_n=y in P*
such that either m*(z\_{i+1},z\_i) <= a or m*(z\_{i+1},z\_i) <= 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 $\aleph\_1$. 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.