In the mathematical study of order, a metric lattice is a lattice that admits a positive valuation: a function satisfying, for any,[1] and
A Boolean algebra is a metric lattice; any finitely-additive measure on its Stone dual gives a valuation.
Every metric lattice is a modular lattice, c.f. lower picture. It is also a metric space, with distance function given by With that metric, the join and meet are uniformly continuous contractions,[2] and so extend to the metric completion (metric space). That lattice is usually not the Dedekind-MacNeille completion, but it is conditionally complete.
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. 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.