In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d ( x , z ) ≤ max { d ( x , y ) , d ( y , z ) } {\displaystyle d(x,z)\leq \max \left\{d(x,y),d(y,z)\right\}} for all x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} . Sometimes the associated metric is also called a non-Archimedean metric or super-metric.
Formal definition An ultrametric on a set M {\displaystyle M} is a real-valued function
d : M × M → R {\displaystyle d\colon M\times M\rightarrow \mathbb {R} }
(where R {\displaystyle \mathbb {R} } denotes the real numbers), such that for all x , y , z ∈ M {\displaystyle x,y,z\in M} :
d ( x , y ) ≥ 0 {\displaystyle d(x,y)\geq 0} with equality if and only if x = y {\displaystyle x=y} (non-degeneracy)
d ( x , y ) = d ( y , x ) {\displaystyle d(x,y)=d(y,x)} (symmetry)
d ( x , z ) ≤ max { d ( x , y ) , d ( y , z ) } {\displaystyle d(x,z)\leq \max\{d(x,y),d(y,z)\}} (strong triangle inequality or ultrametric inequality). An ultrametric space is a pair ( M , d ) {\displaystyle (M,d)} consisting of a set M {\displaystyle M} together with an ultrametric d {\displaystyle d} on M {\displaystyle M} , which is called the space's associated distance function (also called a metric). If d {\displaystyle d} satisfies all of the conditions but the "if and only if" in condition 1 is weakened to "if", then d {\displaystyle d} is called an ultrapseudometric on M {\displaystyle M} . An ultrapseudometric space is a pair ( M , d ) {\displaystyle (M,d)} consisting of a set M {\displaystyle M} and an ultrapseudometric d {\displaystyle d} on M {\displaystyle M} . In the case when M {\displaystyle M} is an Abelian group (written additively) and d {\displaystyle d} is generated by a length function ‖ ⋅ ‖ {\displaystyle \|\cdot \|} (so that d ( x , y ) = ‖ x − y ‖ {\displaystyle d(x,y)=\|x-y\|} ), the last property can be made stronger using the Krull sharpening to:
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