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Algebra and Discrete Mathematics, 2013, том 16, выпуск 1, страницы 107–115
(Mi adm439)
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Эта публикация цитируется в 1 научной статье (всего в 1 статье)
RESEARCH ARTICLE
Ideals in $(\mathcal{Z}^{+},\leq_{D})$
Sankar Sagi Assistant Professor of Mathematics, College of Applied Sciences, Sohar, Sultanate of Oman
Аннотация:
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.
Ключевые слова:
Partial Order, Lattice, Semi Lattice, Convolution, Ideal.
Поступила в редакцию: 17.12.2011 Исправленный вариант: 27.03.2013
Образец цитирования:
Sankar Sagi, “Ideals in $(\mathcal{Z}^{+},\leq_{D})$”, Algebra Discrete Math., 16:1 (2013), 107–115
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/adm439 https://www.mathnet.ru/rus/adm/v16/i1/p107
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