Since the term is overloaded, a brief definition first. A poset is a set $X$ endowed with a partial order $\le$. Given two elements $a,b \in X$, we can define $x \vee y$ (join) as their least upper bound in $X$, and similarly define $x \wedge y$ (meet)(join)as a greatest lower bound. 

> 
> A lattice is a poset in which any two
>  elements have a unique meet and a
>  unique join.
> 
> 
> 

Lattices (in this form) show up in theoryCS in (briefly) the theory of submodularity (with the subset lattice) and clustering (the partition lattice), as well as in domain theory (which I don't understand too well) and static analysis. 

But I'm interested in applications which use metric structures on lattices. A simple example comes from clustering, where any antimonotone submodular function $f : X \rightarrow R$ (antimonotone means that if $x \le y, f(x) \le f(y)$) induces a metric 
$$d(x,y) = 2f(x \wedge y) - f(x) - f(y)$$

This metric has been used extensively as a way to compare two different clusterings of a data set. 

Are there other applications of lattices that care about metric structures ? I've been interested in the domain theory/static analysis application, but so far I've not seen any need for *metrics*.