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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| Date: | 2018 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2018
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/80 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | 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. |
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