Metric lattice explained

In the mathematical study of order, a metric lattice is a lattice that admits a positive valuation: a function satisfying, for any,[1] v(a)+v(b)=v(a\wedge b)+v(a\vee b) and \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.

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,[2] and so extend to the metric completion (metric space). That lattice is usually not the Dedekind-MacNeille completion, but it is conditionally complete.

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. 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.

Notes and References

  1. Book: Rutherford, Daniel Edwin . 1965 . Introduction to Lattice Theory. Oliver and Boyd. 20–22.
  2. Book: Birkhoff, Garrett. AMS Colloquium Publications 25. Lattice Theory. Garrett Birkhoff. Revised. AMS. New York City. 1948. 2027/iau.31858027322886 . HathiTrust.
  3. Kaburlasos, V. G. (2004). "FINs: Lattice Theoretic Tools for Improving Prediction of Sugar Production From Populations of Measurements." IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 34(2), 1017–1030.