# Constructing a smooth lattice from a discrete one.

I have the standard lattice L defined over partitions of $1\ldots n$ under the split-merge relation. I also have an antimonotone function from L to R that's submodular, and so gives me a metric on L via the construction $d(x,y) = 2f(x \wedge y) - f(x)-f(y)$.

In general, but on this lattice specifically, I'm wondering if there's a construction that allows me to define a continuous lattice C with the property that L behaves like the "integer points" on C, and a metric $d'$ can be induced on C that when restricted to L behaves like $d$.

For example, if I have two points $x \le y$ in $L$, I'd like to construct a point $z$ "halfway" between $x$ and $y$ such that under the metric $d', d'(x,z) = d'(z,y)$, and under the ordering that defines $C, x \le z \le y$.

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Thanks ! What is a Dedekind/MacNeille completion ? In response to your suggestion, replacing the line by a copy of [0,1] makes sense, but this only gives me a "skeleton". For example, if I have a diamond in the Hasse diagram consisting of $a$ below b and c, which are below d, I'd like to access the "interior" of this diamond as well. – Suresh Venkat Feb 4 '10 at 8:53