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Ultrametric


An ultrametric is a metric which satisfies the following strengthened version of the triangle inequality,

 d(x,z)<=max(d(x,y),d(y,z))

for all x,y,z. At least two of d(x,y), d(y,z), and d(x,z) are the same.

Let X be a set, and let X^N (where N is the set of natural numbers) denote the collection of sequences of elements of X (i.e., all the possible sequences x_1, x_2, x_3, ...). For sequences a=(a_1,a_2,...), b=(b_1,b_2,...), let n be the number of initial places where the sequences agree, i.e., a_1=b_1, a_2=b_2, ..., a_n=b_n, but a_(n+1)!=b_(n+1). Take n=0 if a_1!=b_1. Then defining d(a,b)=2^(-n) gives an ultrametric.

The p-adic norm metric is another example of an ultrametric.


See also

Metric, p-adic Number

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Cite this as:

Weisstein, Eric W. "Ultrametric." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ultrametric.html

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