Combinatorial characterization of pseudometrics

  title={Combinatorial characterization of pseudometrics},
  author={Oleksiy Dovgoshey and Juoni Luukkainen},
  journal={Acta Mathematica Hungarica},
Let X, Y be sets and let $$\Phi, \Psi$$ Φ , Ψ be mappings with the domains X 2 and Y 2 respectively. We say that $$\Phi$$ Φ is combinatorially similar to $$\Psi$$ Ψ if there are bijections $$f \colon \Phi(X^2) \to \Psi(Y^{2})$$ f : Φ ( X 2 ) → Ψ ( Y 2 ) and $$g \colon Y \to X$$ g : Y → X such that $$\Psi(x, y) = f(\Phi(g(x), g(y)))$$ Ψ ( x , y ) = f ( Φ ( g ( x ) , g ( y ) ) ) for all $$x, y \in Y$$ x , y ∈ Y . It is shown that the semigroups of binary relations generated by sets $$\{\Phi^{-1… 
Combinatorial properties of ultrametrics and generalized ultrametrics
  • O. Dovgoshey
  • Mathematics
    Bulletin of the Belgian Mathematical Society - Simon Stevin
  • 2020
Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ and $\Psi$ are combinatorially similar if there are bijections $f \colon
Ultrametric Preserving Functions and Weak Similarities of Ultrametric Spaces$$^*$$
Let $WS(X, d)$ be the class of ultrametric spaces which are weakly similar to ultrametric space $(X, d)$. The main results of the paper completely describe the ultrametric spaces $(X, d)$ for which
When all Permutations are Combinatorial Similarities
. Let ( X, d ) be a semimetric space. A permutation Φ of the set X is a combinatorial self similarity of ( X, d ) if there is a bijective function f : d ( X 2 ) → d ( X 2 ) such that d ( x, y ) = f (
Completeness, Closedness and Metric Reflections of Pseudometric Spaces
. It is well-known that a metric space ( X, d ) is complete iff the set X is closed in every metric superspace of ( X, d ) . For a given pseudometric space ( Y, ρ ) , we describe the maximal class CEC
Ultrametrics and Complete Multipartite Graphs
We describe the class of graphs for which all metric spaces with diametrical graphs belonging to this class are ultrametric. It is shown that a metric space (X, d) is ultrametric iff the diametrical
Let (X, d) be an unbounded metric space. To investigate the asymptotic behavior of (X, d) at infinity, one can consider a sequence of rescaling metric spaces (X, 1 rn d) generated by given sequence
On equivalence of unbounded metric spaces at infinity
Let (X, d) be an unbounded metric space. To investigate the asymptotic behavior of (X, d) at infinity, one can consider a sequence of rescaling metric spaces (X, 1 rn d) generated by given sequence
Uniqueness of best proximity pairs and rigidity of semimetric spaces
For arbitrary semimetric space ( X, d ) and disjoint proximinal subsets A , B of X we define the proximinal graph as a bipartite graph with parts A and B whose edges { a, b } satisfy the equality d (


  • O. Dovgoshey
  • Mathematics
    International Electronic Journal of Algebra
  • 2019
Let $X$ be a nonempty set and $X^{2}$ be the Cartesian square of $X$. Some semigroups of binary relations generated partitions of $X^2$ are studied. In particular, the algebraic structure of
Strongly rigid metrics and zero dimensionality
A metric d is strongly rigid if and only if d(x, y) =# d(w, z) whenever the doubleton (x,y} is not equal to the doubleton {w, z}. It is shown that a nonempty metrizable space X admits a compatible
Weak similarities of metric and semimetric spaces
Let (X,dX) and (Y,dY) be semimetric spaces with distance sets D(X) and D(Y), respectively. A mapping F: X→Y is a weak similarity if it is surjective and there exists a strictly increasing f:
A remark on M. M. Day’s characterization of inner-product spaces and a conjecture of L. M. Blumenthal
1. A space of elements a, b, ■ • • , with a distance function ab is said to be semi-metric provided ab = ba>0 if a^b, and aa = 0. A reallinear space of elements/, g, ■ • ■ is said to be semi-normed
On Metric Arcs of Vanishing Menger Curvature
1. Let r be a metric space which is a simple arc, that is the topological image of a closed linear segment. Menger introduced the following purely metric definition of curvature ([6], pp. 480, 481).2
The algebraic theory of semigroups
This book, along with volume I, which appeared previously, presents a survey of the structure and representation theory of semi groups. Volume II goes more deeply than was possible in volume I into
The theory of partitions
1. The elementary theory of partitions 2. Infinite series generating functions 3. Restricted partitions and permutations 4. Compositions and Simon Newcomb's problem 5. The Hardy-Ramanujan-Rademacher
A metric characterization of zero-dimensional spaces
It is shown that a nonempty separable metrizable space X is zero-dimensional if and only if there exists a metric p on X, inducing the given topology of X and such that all nonzero distances p(x, y)
Day’s characterization of inner-product spaces and a conjecture of L
  • Proc. Amer. Math. Soc
  • 1952
Undergraduate Algebra
  • M. Brešar
  • Mathematics
    Springer Undergraduate Mathematics Series
  • 2019