Metric lattice

In the mathematical study of order, a metric lattice L is a lattice that admits a positive valuation: a function v ∈ L → ℝ satisfying, for any a, b ∈ L,^[1] $${\displaystyle v(a)+v(b)=v(a\wedge b)+v(a\vee b)}$$ and $${\displaystyle {a>b}\Rightarrow v(a)>v(b){\text{.}}}$$