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]
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]
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