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

Example valuation function on the cube lattice which makes it a metric lattice.

Relation to other notions

A lattice containing N5 (depicted) cannot be a metric one, since v(d)+v(c) = v(e)+v(a) = v(b)+v(c) implies v(d) = v(b), contradicting v(d) < v(b).

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

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}