Metric lattice

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 $v \in L \to ℝ$ satisfying, for any $a, b \in L$ ,^[1]

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.^[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]

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}

References

1 2 Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd. pp. 20–22.
1 2 3 4 5 Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust.
1 2 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

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[LatticeThry1-1] 1 2 Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd. pp. 20–22.

[LatticeThry2-2] 1 2 3 4 5 Birkhoff, Garrett (1948). Lattice Theory. AMS Colloquium Publications 25 (Revised ed.). New York City: AMS. hdl:2027/iau.31858027322886 – via HathiTrust.

[FuzzyLogic-3] 1 2 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

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