Characterization of regular convolutions
A convolution is a mapping \(\mathcal{C}\) of the set \(Z^{+}\) of positive integers into the set \({\mathscr{P}}(Z^{+})\) of all subsets of \(Z^{+}\) such that, for any \(n\in Z^{+}\), each member of \(\mathcal{C}(n)\) is a divisor of \(n\). If \(\mathcal{D}(n)\) is the set of all divisors of \(n\)...
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| Дата: | 2018 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/80 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543166180622336 |
|---|---|
| author | Sagi, Sankar |
| author_facet | Sagi, Sankar |
| author_sort | Sagi, Sankar |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-05-17T07:54:05Z |
| description | A convolution is a mapping \(\mathcal{C}\) of the set \(Z^{+}\) of positive integers into the set \({\mathscr{P}}(Z^{+})\) of all subsets of \(Z^{+}\) such that, for any \(n\in Z^{+}\), each member of \(\mathcal{C}(n)\) is a divisor of \(n\). If \(\mathcal{D}(n)\) is the set of all divisors of \(n\), for any \(n\), then \(\mathcal{D}\) is called the Dirichlet's convolution [2]. If \(\mathcal{U}(n)\) is the set of all Unitary(square free) divisors of \(n\), for any \(n\), then \(\mathcal{U}\) is called unitary(square free) convolution. Corresponding to any general convolution \(\mathcal{C}\), we can define a binary relation \(\leq_{\mathcal{C}}\) on \(Z^{+}\) by `\(m\leq_{\mathcal{C}}n\) if and only if \( m\in \mathcal{C}(n)\)'. In this paper, we present a characterization of regular convolution. |
| first_indexed | 2026-02-08T07:58:06Z |
| format | Article |
| id | admjournalluguniveduua-article-80 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:58:06Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-802018-05-17T07:54:05Z Characterization of regular convolutions Sagi, Sankar semilattice, lattice, convolution, multiplicative, co-maximal, prime filter, cover, regular convolution 06B10,11A99 A convolution is a mapping \(\mathcal{C}\) of the set \(Z^{+}\) of positive integers into the set \({\mathscr{P}}(Z^{+})\) of all subsets of \(Z^{+}\) such that, for any \(n\in Z^{+}\), each member of \(\mathcal{C}(n)\) is a divisor of \(n\). If \(\mathcal{D}(n)\) is the set of all divisors of \(n\), for any \(n\), then \(\mathcal{D}\) is called the Dirichlet's convolution [2]. If \(\mathcal{U}(n)\) is the set of all Unitary(square free) divisors of \(n\), for any \(n\), then \(\mathcal{U}\) is called unitary(square free) convolution. Corresponding to any general convolution \(\mathcal{C}\), we can define a binary relation \(\leq_{\mathcal{C}}\) on \(Z^{+}\) by `\(m\leq_{\mathcal{C}}n\) if and only if \( m\in \mathcal{C}(n)\)'. In this paper, we present a characterization of regular convolution. Lugansk National Taras Shevchenko University 2018-04-27 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/80 Algebra and Discrete Mathematics; Vol 25, No 1 (2018) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/80/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/80/119 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/80/120 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | semilattice lattice convolution multiplicative co-maximal prime filter cover regular convolution 06B10,11A99 Sagi, Sankar Characterization of regular convolutions |
| title | Characterization of regular convolutions |
| title_full | Characterization of regular convolutions |
| title_fullStr | Characterization of regular convolutions |
| title_full_unstemmed | Characterization of regular convolutions |
| title_short | Characterization of regular convolutions |
| title_sort | characterization of regular convolutions |
| topic | semilattice lattice convolution multiplicative co-maximal prime filter cover regular convolution 06B10,11A99 |
| topic_facet | semilattice lattice convolution multiplicative co-maximal prime filter cover regular convolution 06B10,11A99 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/80 |
| work_keys_str_mv | AT sagisankar characterizationofregularconvolutions |