Ultrametric space

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) ight}} . Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications. 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) ight}} . Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications. Formally, an ultrametric space is a set of points M {displaystyle M} with an associated distance function (also called a metric) (where R {displaystyle mathbb {R} } is the set of real numbers), such that for all x , y , z ∈ M {displaystyle x,y,zin M} , one has: In the case when M {displaystyle M} is a 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: We want to prove that if ‖ x + y ‖ ≤ max { ‖ x ‖ , ‖ y ‖ } {displaystyle |x+y|leq max left{|x|,|y| ight}} , then the equality occurs if ‖ x ‖ ≠ ‖ y ‖ {displaystyle |x| eq |y|} . Without loss of generality, let us assume that ‖ x ‖ > ‖ y ‖ {displaystyle |x|>|y|} . This implies that ‖ x + y ‖ ≤ ‖ x ‖ {displaystyle |x+y|leq |x|} . But we can also compute ‖ x ‖ = ‖ ( x + y ) − y ‖ ≤ max { ‖ x + y ‖ , ‖ y ‖ } {displaystyle |x|=|(x+y)-y|leq max left{|x+y|,|y| ight}} . Now, the value of max { ‖ x + y ‖ , ‖ y ‖ } {displaystyle max left{|x+y|,|y| ight}} cannot be ‖ y ‖ {displaystyle |y|} , for if that is the case, we have ‖ x ‖ ≤ ‖ y ‖ {displaystyle |x|leq |y|} contrary to the initial assumption. Thus, max { ‖ x + y ‖ , ‖ y ‖ } = ‖ x + y ‖ {displaystyle max left{|x+y|,|y| ight}=|x+y|} , and ‖ x ‖ ≤ ‖ x + y ‖ {displaystyle |x|leq |x+y|} . Using the initial inequality, we have ‖ x ‖ ≤ ‖ x + y ‖ ≤ ‖ x ‖ {displaystyle |x|leq |x+y|leq |x|} and therefore ‖ x + y ‖ = ‖ x ‖ {displaystyle |x+y|=|x|} . From the above definition, one can conclude several typical properties of ultrametrics. For example, in an ultrametric space, for all x , y , z ∈ M {displaystyle x,y,zin M} and r , s ∈ R {displaystyle r,sin mathbb {R} } , at least one of the three equalities d ( x , y ) = d ( y , z ) {displaystyle d(x,y)=d(y,z)} or d ( x , z ) = d ( y , z ) {displaystyle d(x,z)=d(y,z)} or d ( x , y ) = d ( z , x ) {displaystyle d(x,y)=d(z,x)} holds. That is, every triple of points in the space forms an isosceles triangle, so the whole space is an isosceles set. In the following, the concept and notation of an (open) ball is the same as in the article about metric spaces, i.e. 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.