Metric lattice

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

In the mathematical study of order, a metric lattice Template:Mvar is a lattice that admits a positive valuation: a function Template:Math satisfying, for any Template:Math, $$v(a)+v(b)=v(a\wedge b)+v(a\vee b)$$ and $$a>b\Rightarrow v(a)>v(b)\text{.}$$

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.Template:Rp

Every metric lattice is a modular lattice, c.f. lower picture. It is also a metric space, with distance function given by $$d(x,y)=v(x\vee y)-v(x\wedge y)\text{.}$$ With that metric, the join and meet are uniformly continuous contractions,Template:Rp and so extend to the metric completion (metric space). That lattice is usually not the Dedekind-MacNeille completion, but it is conditionally complete.Template:Rp

Applications

In the study of fuzzy logic and interval arithmetic, the space of uniform distributions is a metric lattice. Metric lattices are also key to von Neumann's construction of the continuous projective geometry.Template:Rp 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.Template:Rp