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] and
A Boolean algebra is a metric lattice; any finitely-additive measure on its Stone dual gives a valuation. [2]: 252–254
Every metric lattice is a modular lattice, [1] c.f. lower picture. It is also a metric space, with distance function given by [3] With that metric, the join and meet are uniformly continuous contractions, [2]: 77 and so extend to the metric completion (metric space). That lattice is usually not the Dedekind-MacNeille completion, but it is conditionally complete. [2]: 80
In the study of fuzzy logic and interval arithmetic, the space of uniform distributions is a metric lattice. [3] Metric lattices are also key to von Neumann's construction of the continuous projective geometry. [2]: 126 A function satisfies the one-dimensional wave equation if and only if it is a valuation for the lattice of spacetime coordinates with the natural partial order. A similar result should apply to any partial differential equation solvable by the method of characteristics, but key features of the theory are lacking. [2]: 150–151