$Ideals in $(\mathcal{Z}^{+},\leq_{D})$$

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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Date:	2018
Main Author:	Sagi, Sankar
Format:	Article
Language:	English
Published:	Lugansk National Taras Shevchenko University 2018
Subjects:	Partial Order,Lattice,Semi Lattice,Convolution,Ideal 06B10,11A99
Online Access:	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/760
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Journal Title:	Algebra and Discrete Mathematics

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Algebra and Discrete Mathematics

id	admjournalluguniveduua-article-760
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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
institution	Algebra and Discrete Mathematics
baseUrl_str
datestamp_date	2018-04-26T01:26:05Z
collection	OJS
language	English
topic	Partial Order,Lattice,Semi Lattice,Convolution,Ideal 06B10,11A99
spellingShingle	Partial Order,Lattice,Semi Lattice,Convolution,Ideal 06B10,11A99 Sagi, Sankar Ideals in $(\mathcal{Z}^{+},\leq_{D})$
topic_facet	Partial Order,Lattice,Semi Lattice,Convolution,Ideal 06B10,11A99
format	Article
author	Sagi, Sankar
author_facet	Sagi, Sankar
author_sort	Sagi, Sankar
title	Ideals in $(\mathcal{Z}^{+},\leq_{D})$
title_short	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_sort	ideals in $(\mathcal{z}^{+},\leq_{d})$
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.
publisher	Lugansk National Taras Shevchenko University
publishDate	2018
url	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/760
work_keys_str_mv	AT sagisankar idealsinmathcalzleqd
first_indexed	2025-12-02T15:44:54Z
last_indexed	2025-12-02T15:44:54Z
_version_	1850411890832834560

Ideals in \((\mathcal{Z}^{+},\leq_{D})\)

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