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]

$${\displaystyle v(a)+v(b)=v(a\wedge b)+v(a\vee b)}$$

$${\displaystyle {a>b}\Rightarrow v(a)>v(b){\text{.))}$$

Relation to other notions

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]

$${\displaystyle d(x,y)=v(x\vee y)-v(x\wedge y){\text{.))}$$

Applications

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}