Ideals in \((\mathcal{Z}^{+},\leq_{D})\)
A convolution is a mapping \(\mathcal{C}\) of the set \(\mathcal{Z}^{+}\) of positive integers into the set \(\mathcal{P}(\mathcal{Z}^{+})\) of all subsets of \(\mathcal{Z}^{+}\) such that every member of \(\mathcal{C}(n)\) is a divisor of \(n\). If for any \(n\), \(D(n)\) is the set of all positiv...
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| Datum: | 2018 |
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Lugansk National Taras Shevchenko University
2018
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/760 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543316410105856 |
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| author | Sagi, Sankar |
| author_facet | Sagi, Sankar |
| author_sort | Sagi, Sankar |
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| collection | OJS |
| datestamp_date | 2018-04-26T01:26:05Z |
| description | A convolution is a mapping \(\mathcal{C}\) of the set \(\mathcal{Z}^{+}\) of positive integers into the set \(\mathcal{P}(\mathcal{Z}^{+})\) of all subsets of \(\mathcal{Z}^{+}\) such that every member of \(\mathcal{C}(n)\) is a divisor of \(n\). If for any \(n\), \(D(n)\) is the set of all positive divisors of \(n\) , then \(D\) is called the Dirichlet's convolution. It is well known that \(\mathcal{Z}^{+}\) has the structure of a distributive lattice with respect to the division order. Corresponding to any general convolution \(\mathcal{C}\), one can define a binary relation \(\leq_{\mathcal{C}}\) on \(\mathcal{Z}^{+}\) by ` \(m\leq_{\mathcal{C}}n \) if and only if \( m\in \mathcal{C}(n)\) ' . A general convolution may not induce a lattice on \(\mathcal{Z^{+}}\) . However most of the convolutions induce a meet semi lattice structure on \(\mathcal{Z^{+}}\) .In this paper we consider a general meet semi lattice and study it's ideals and extend these to \((\mathcal{Z}^{+},\leq_{D})\) , where \(D\) is the Dirichlet's convolution. |
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| format | Article |
| id | admjournalluguniveduua-article-760 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:44:54Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
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| spelling | admjournalluguniveduua-article-7602018-04-26T01:26:05Z Ideals in \((\mathcal{Z}^{+},\leq_{D})\) Sagi, Sankar Partial Order,Lattice,Semi Lattice,Convolution,Ideal 06B10,11A99 A convolution is a mapping \(\mathcal{C}\) of the set \(\mathcal{Z}^{+}\) of positive integers into the set \(\mathcal{P}(\mathcal{Z}^{+})\) of all subsets of \(\mathcal{Z}^{+}\) such that every member of \(\mathcal{C}(n)\) is a divisor of \(n\). If for any \(n\), \(D(n)\) is the set of all positive divisors of \(n\) , then \(D\) is called the Dirichlet's convolution. It is well known that \(\mathcal{Z}^{+}\) has the structure of a distributive lattice with respect to the division order. Corresponding to any general convolution \(\mathcal{C}\), one can define a binary relation \(\leq_{\mathcal{C}}\) on \(\mathcal{Z}^{+}\) by ` \(m\leq_{\mathcal{C}}n \) if and only if \( m\in \mathcal{C}(n)\) ' . A general convolution may not induce a lattice on \(\mathcal{Z^{+}}\) . However most of the convolutions induce a meet semi lattice structure on \(\mathcal{Z^{+}}\) .In this paper we consider a general meet semi lattice and study it's ideals and extend these to \((\mathcal{Z}^{+},\leq_{D})\) , where \(D\) is the Dirichlet's convolution. Lugansk National Taras Shevchenko University 2018-04-26 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/760 Algebra and Discrete Mathematics; Vol 16, No 1 (2013) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/760/289 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | Partial Order,Lattice,Semi Lattice,Convolution,Ideal 06B10,11A99 Sagi, Sankar Ideals in \((\mathcal{Z}^{+},\leq_{D})\) |
| title | Ideals in \((\mathcal{Z}^{+},\leq_{D})\) |
| title_full | Ideals in \((\mathcal{Z}^{+},\leq_{D})\) |
| title_fullStr | Ideals in \((\mathcal{Z}^{+},\leq_{D})\) |
| title_full_unstemmed | Ideals in \((\mathcal{Z}^{+},\leq_{D})\) |
| title_short | Ideals in \((\mathcal{Z}^{+},\leq_{D})\) |
| title_sort | ideals in \((\mathcal{z}^{+},\leq_{d})\) |
| topic | Partial Order,Lattice,Semi Lattice,Convolution,Ideal 06B10,11A99 |
| topic_facet | Partial Order,Lattice,Semi Lattice,Convolution,Ideal 06B10,11A99 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/760 |
| work_keys_str_mv | AT sagisankar idealsinmathcalzleqd |