Vector-valued valuations on lattices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:29:12Z http://mathoverflow.net/feeds/question/27058 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27058/vector-valued-valuations-on-lattices Vector-valued valuations on lattices Suresh Venkat 2010-06-04T15:56:13Z 2010-10-10T06:06:18Z <p>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 $$v(x) + v(y) = v(x \wedge y) + v(x \vee y)$$</p> <p>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) ? </p> <p>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)</p> <p>This is a reference request, for the most part. </p>