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Metric lattice

From Wikipedia, the free encyclopedia
Example valuation function on the cube lattice which makes it a metric lattice.

In the mathematical study of order, a metric lattice L is a lattice that admits a positive valuation: a function vL → ℝ satisfying, for any a, bL,[1] and

Relation to other notions

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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] 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

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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 

References

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  1. ^ a b Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd. pp. 20–22.
  2. ^ a b c d e Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust.
  3. ^ a b 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. doi:10.1109/tsmcb.2003.818558