Linear isometries on Weighted Coordinates Poset Block Space

Given \([n]=\{1,2,\ldots,n\}\), a poset order \(\preceq\) on \([n]\), a label map \(\pi : [n] \rightarrow \mathbb{N}\) defined by \(\pi(i)=k_i\) with \(\sum_{i=1}^{n}\pi (i) = N\), and a weight function \(w\) on \(\mathbb{F}_{q}\), let \(\mathbb{F}_{q}^N\) be the vector space of \(N\)-tuples over th...

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Bibliographic Details
Published inarXiv.org
Main Authors Shriwastva, Atul Kumar, Selvaraj, R S
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 17.11.2022
Subjects
Metric space
Set theory
Vector spaces
Weighting functions
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Summary:Given \([n]=\{1,2,\ldots,n\}\), a poset order \(\preceq\) on \([n]\), a label map \(\pi : [n] \rightarrow \mathbb{N}\) defined by \(\pi(i)=k_i\) with \(\sum_{i=1}^{n}\pi (i) = N\), and a weight function \(w\) on \(\mathbb{F}_{q}\), let \(\mathbb{F}_{q}^N\) be the vector space of \(N\)-tuples over the field \(\mathbb{F}_{q}\) equipped with \((P,w,\pi)\)-metric where \( \mathbb{F}_q^N \) is the direct sum of spaces \( \mathbb{F}_{q}^{k_1}, \mathbb{F}_{q}^{k_2}, \ldots, \mathbb{F}_{q}^{k_n} \). In this paper, we determine the groups of linear isometries of \((P,w,\pi)\)-metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset (block) metric spaces. In particular, we re-obtain the group of linear isometries of the \((P,w)\)-mertic spaces and \((P,\pi)\)-mertic spaces.
ISSN:2331-8422