In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to for all x, y, and z. Sometimes the associated metric is also called a non-Archimedean metric or super-metric.

Formal definition

An ultrametric on a set M is a real-valued function (where ℝ denote the real numbers), such that for all x, y, z ∈ M d(x, y) ≥ 0 d(x, y) = d(y, x) (symmetry); d(x, x) = 0 d(x, y) = 0 then x = y d(x, z) ≤ max {d(x, y), d(y, z) } (strong triangle inequality or ultrametric inequality). An ultrametric space is a pair (M, d) consisting of a set M together with an ultrametric d on M, which is called the space's associated distance function (also called a metric). If d satisfies all of the conditions except possibly condition 4 then d is called an ultrapseudometric on M. An ultrapseudometric space is a pair (M, d) consisting of a set M and an ultrapseudometric d on M. In the case when M is an Abelian group (written additively) and d is generated by a length function |\cdot| (so that ), the last property can be made stronger using the Krull sharpening to: We want to prove that if, then the equality occurs if. Without loss of generality, let us assume that This implies that. But we can also compute. Now, the value of cannot be |y|, for if that is the case, we have contrary to the initial assumption. Thus,, and. Using the initial inequality, we have and therefore.

Properties

From the above definition, one can conclude several typical properties of ultrametrics. For example, for all x,y,z \in M, at least one of the three equalities or or holds. That is, every triple of points in the space forms an isosceles triangle, so the whole space is an isosceles set. Defining the (open) ball of radius r > 0 centred at x \in M as, we have the following properties: Proving these statements is an instructive exercise. All directly derive from the ultrametric triangle inequality. Note that, by the second statement, a ball may have several center points that have non-zero distance. The intuition behind such seemingly strange effects is that, due to the strong triangle inequality, distances in ultrametrics do not add up.

Examples

Applications