Monday, 2 February 2009

mg.metric geometry - Constructing a metric over a lattice

Consider a lattice (calL,wedge,vee) with an antimonotonic function f:calLrightarrowmathbbR defined on it (i.e xpreceqyimpliesf(x)gef(y)).



f is said to be submodular if for all x,yincalL, f(x)+f(y)gef(xwedgey)+f(xveey)

and supermodular if the inequality is flipped (again for all x,y).



It's generally known (there's an easy proof), that a submodular f induces a metric on calL via the defn ds(x,y)=2f(xwedgey)f(x)f(y)

. If f is supermodular, then the construction ds(x,y)=f(x)+f(y)2f(xveey)
yields a metric.



Question I'm dealing with an f that is nether sub- nor supermodular. I can define the "distance" d(x,y)=min(ds(x,y),ds(x,y))



Conjecture: d(x,y) is a metric.



I have very little sound mathematical intuition for why this conjecture should be true, and bucketloads of empirical evidence (from a lattice I'm actually working with). This seems like the kind of thing that if true, would be reasonably well known to experts, and if false, might have a clear counterexample. So this is a plea for help.



Since it might make a difference, I should mention that the lattice I'm working with is nondistributive in general, but it has distributive sublattices where I'm still unable to prove the conjecture.

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