On ultrametric-preserving functions

Abstract

Characterizations of pseudoultrametric-preserving functions and semimetricpreserving functions are found. The structural properties of pseudoultrametrics which can be represented as a composition of an ultrametric and ultrametric-pseudoultrametric-preserving function are found. A dual form of Pongsriiam-Termwuttipong characterization of the ultrametricpreserving functions is described. We also introduce a concept of k-separating family of functions and use it to characterize the ultrametric spaces.2010 Mathematics Subject Classification. 54E35.

P-adic metric preserving functions and their analogues

“…Let (i) hold and let (X, d) ∈ U be given. To prove that f • d is an ultrametric it suffices to show that the strong triangle inequality (13) f (d(x, y)) ≤ max{f (d(x, z)), f (d(z, y))} holds for all x, y, z ∈ X. Since (X, d) is an ultrametric space, there is a permutation ( x y z x 1 x 2 x 3 ) such that d(x 1 , x 2 ) = d(x 2 , x 3 ) holds.…”

On ultrametrics, b-metrics, w-distances, metric-preserving functions, and fixed point theorems

“…These distances and generalized metrics also have many applications. Examples and applications of ultrametrics can be found in [2,5,6,14,15,18,19,21,26,28]. We also refer the reader to [8,9,12,20,[22][23][24] for examples and applications of w-distances.…”

On Functions Preserving Products of Certain Classes of Semimetric Spaces

“…The idea can be traced back to Wilson [51] and Sreenivasan [43]. The research on this subject was later conducted by Borsík, Doboš and Piotrowski (see [3][4][5][6][10][11][12][13][14]), Corazza (see [7]), Das (see [9]), Dovgoshey (see [15][16][17]49]), Jůza (see [25]), Khemaratchatakumthorn, Pongsriiam, Termwuttipong and Samphavat (see [27-29, 40-42, 44]), Pokorny (see [37][38][39]), Vallin (see [47][48][49]), and recently, also by Jachymski and Turoboś (see [22,46]).…”