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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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Here is a suggestion: Try "realizing" the Hasse diagram of the lattice. (Pun intended.) Draw the Hasse diagram of the underlying poset, and replace each cover in the diagram (line which indicates x > y and no element of the lattice lies between) by a copy of the real interval (0,1). I do not know if the submodularity of the d function will be preserved, but it may be a step in the direction of what you want.

More generally this realization step with covers, combined with some form of Dedekind MacNeille completion may work, if the original lattice has some intervals in which covers do not exist.

Gerhard "Ask Me About System Design" Paseman, 2010.02.04

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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
@Suresh: If you want the interior, and so on, you maybe want something like the order complex of the poset, or the geometrical realization. I do not think you can turn these into posets in any significant way, though. – Mariano Suárez-Alvarez Feb 4 '10 at 17:37
Like Dedekind completed the linear order of the rationals using downsets to make reals, so one can do so for lattices to turn a lattice into a complete lattice. This is Dedekind-MacNeille completion. Also, see if you can provide satisfactory examples on M3, N5, and lattices which are incidence relations of small projective spaces. If you know what your result looks like for those examples, then maybe an appropriate generalization can be found for other finite lattices. (E.g., for the Fano plane, draw 7 line dots above 7 point dotss, each line covers the three points it contains.) – Gerhard Paseman Feb 4 '10 at 23:39

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