Combinatorial characterization of pseudometrics
@article{Dovgoshey2020CombinatorialCO, title={Combinatorial characterization of pseudometrics}, author={Oleksiy Dovgoshey and Juoni Luukkainen}, journal={Acta Mathematica Hungarica}, year={2020}, volume={161}, pages={257-291} }
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 $$\{\Phi^{-1…
8 Citations
Combinatorial properties of ultrametrics and generalized ultrametrics
- MathematicsBulletin of the Belgian Mathematical Society - Simon Stevin
- 2020
Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ and $\Psi$ are combinatorially similar if there are bijections $f \colon…
Ultrametric Preserving Functions and Weak Similarities of Ultrametric Spaces$$^*$$
- Mathematicsp-Adic Numbers, Ultrametric Analysis and Applications
- 2021
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
- Mathematics
- 2022
. 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
- Mathematics
- 2022
. 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
- MathematicsTheory and Applications of Graphs
- 2022
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…
ON THE ASYMPTOTIC EQUIVALENCE OF UNBOUNDED METRIC SPACES
- Mathematics
- 2021
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
- Mathematics
- 2021
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
- Mathematics
- 2022
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 (…
References
SHOWING 1-10 OF 14 REFERENCES
SEMIGROUPS GENERATED BY PARTITIONS
- MathematicsInternational 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
- Mathematics
- 1977
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
- Mathematics
- 2012
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
- Mathematics
- 1952
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
- Mathematics
- 1940
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
- Mathematics
- 1964
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
- Mathematics
- 1976
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
- Mathematics
- 1972
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