Metric and topology on the poset of compact pseudoultrametrics

Nykorovych, Svyatoslav; Nykyforchyn, Oleh

doi:10.15330/cmp.15.2.321-330

Cited by 2 publications

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“…More on metric and topology on posets of compact ultrapseudometrics, in particular, on an alternative method of metrization, is said in our paper [10].…”

Section: Discussion and Future Researchmentioning

confidence: 99%

“…This paper serves two purposes: it summarizes the previous research [8][9][10] and presents new results that finally allow to get a more or less complete view what the posets in question are.…”

Section: Introductionmentioning

confidence: 99%

“…Section 6 contains the central result of the paper, which is reflected in its title. It relies partially on [10] and shows that the approximations from below and from above "play well together", hence they determines the same Lawson topology [2] on the poset of all compact ultrapseudometrics not exceeding a fixed one.…”

Section: Introductionmentioning

confidence: 99%

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Linked bicontinuity of posets of compact ultrapseudometrics

Nykyforchyn

Nykorovych

2023

Preprint

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Approximation relations (``way below'' and ``way above'') onpointwise ordered sets of ultrapseudometrics are studied. It isproved that, to obtain continuity and/or dual continuity of theseposets in the sense considered in domain theory, i.e., possibilityto ``safely'' approximate all elements from below and from aboverespectively, one should restrict to compact ultrapseudometrics,not exceeding a fixed one. Such poset is a complete lattice,the partial order on it determines the Lawson topology and the dualLawson topology, which coincide (which is linked bicontinuity), arecompact Hausdorff, and agree with the uniform convergence metric.Necessary and sufficient conditions are proved for ``way below''and ``way above'' relations to hold. MSC Classification: 06F30 , 54E35

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“…More on metric and topology on posets of compact ultrapseudometrics, in particular, on an alternative method of metrization, is said in our paper [10].…”

Section: Discussion and Future Researchmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linked bicontinuity of posets of compact ultrapseudometrics

Nykyforchyn

Nykorovych

2023

Preprint

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Symmetric linear functionals on the Banach space generated by pseudometrics

Nykorovych,

Vasylyshyn

2024

Mat. Stud.

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In this work we consider the notion of $B$-equivalence of pseudometrics.Two pseudometrics $d_1$ and $d_2$ on a set $X$ are called $B$-equivalent, where $B$ is a subgroup of the group of all bijections on $X,$ if there exists an element $b$ of $B$ such that $d_1(x,y) = d_2(b(x),b(y))$ for every $x,y\in X,$ that is, $d_1$ can be obtained from $d_2$ by permutating elements of $X$ with the aid of the bijection $b.$The group $B$ generates the group $\widehat B$ of transformations of the set of all pseudometricson $X,$ elements of which act as $d(\cdot, \cdot)\mapsto d(b(\cdot),b(\cdot)),$ where $d$ is a pseudometrics on $X$ and $b\in B.$ A function $f$ on the set of all pseudometrics on $X$is called $\widehat B$-symmetric if $f$ is invariant under the action on its argument of elements of the group $\widehat B.$If two pseudometrics $d_1$ and $d_2$ are $B$-equivalent, then $f(d_1)=f(d_2)$ for every $\widehat B$-symmetric function $f.$ In general, the technique of symmetric functions is well-developed for the case of symmetric continuous polynomials and, in particular, for the case of symmetric continuous linear functionals on Banach spaces. To use this technique for the construction of $\widehat B$-symmetricfunctions on sets of pseudometrics, we map these sets to some appropriate Banach space $V$, which is isometrically isomorphic to the Banach space $\ell_1$of all absolutely summing real sequences. Weinvestigate symmetric (with respect to an arbitrary group of symmetry, elements of whichmap the standard Schauder basis of $\ell_1$ into itself) linear continuous functionalson $\ell_1.$ We obtain the complete description of the structure of these functionals.Also we establish analogical results for symmetric linear continuous functionals on the space $V.$ These results are used for the construction of $\widehat B$-symmetric functionals on the set of all pseudometrics on an arbitrary set $X$ for the following case:the group $B$ of bijections on $X,$ that generates the group $\widehat B,$ is such that the set of all $x\in X,$ for which there exists $b\in B$ such that $b(x)\neq x,$is finite.

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Metric and topology on the poset of compact pseudoultrametrics

Cited by 2 publications

References 0 publications

Linked bicontinuity of posets of compact ultrapseudometrics

Linked bicontinuity of posets of compact ultrapseudometrics

Symmetric linear functionals on the Banach space generated by pseudometrics

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