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There's a fair amount of work on valuations on (modular) lattices, by which I mean functions $v : \mathcal{L} \rightarrow R$ that satisfy the modular expression $$v(x) + v(y) = v(x \wedge y) + v(x \vee y) $$

I'm wondering if there's been any work on vector-valued valuations (where the range of v is $R^k$ and the same relation holds) ?

In addition, I'm also interested in lower valuations (I'm not sure if this name is standard) that satisfy the submodular inequality $$v(x) + v(y) \ge v(x \wedge y) + v(x \vee y) $$ and possibly the generalization to $R^k$ where we replace the above by $$v(x) + v(y) \succeq v(x \wedge y) + v(x \vee y) $$ ($\succeq$ being the coordinate-wise partial order)

This is a reference request, for the most part.

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Is the last inequality $\succeq$ coordinatewise? –  François G. Dorais Jun 4 '10 at 16:07
    
Yes, I was using the coordinate-wise partial order on $R^k$. I'll edit –  Suresh Venkat Jun 4 '10 at 16:50
1  
Such a $v$ is a $k$-ple of real valuations. Usually one studies the real linear combinations of the components, or the compact convex set of all real $v$ with $v(1)=1$, see Goodearl, von Neumann regular rings (rank functions on a regular ring are almost the same thing as normalized valuations on the lattice). In a direct product of $k$ irreducible continuous geometries your get your case. –  user46855 Feb 15 at 15:25
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