Metric Lattice
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In the mathematical study of order, a metric lattice is a lattice that admits a positive valuation: a function satisfying, for any , 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 In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
is a metric lattice; any finitely-additive measure on its Stone dual gives a valuation. Every metric lattice is a
modular lattice In the branch of mathematics called order theory, a modular lattice is a lattice (order), lattice that satisfies the following self-duality (order theory), dual condition, ;Modular law: implies where are arbitrary elements in the lattice, &nbs ...
, c.f. lower picture. It is also a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, 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, 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 Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
and
interval arithmetic Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods using ...
, the space of
uniform distribution Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
s is a metric lattice.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. do
10.1109/tsmcb.2003.818558
/ref> Metric lattices are also key to von Neumann's construction of the continuous projective geometry. A function satisfies the one-dimensional
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
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 In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
solvable by the
method of characteristics In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial d ...
, but key features of the theory are lacking.


References

{{math-stub Lattice theory Metric spaces