$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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Дата:	2018
Автор:	Sagi, Sankar
Формат:	Стаття
Мова:	English
Опубліковано:	Lugansk National Taras Shevchenko University 2018
Теми:	Partial Order,Lattice,Semi Lattice,Convolution,Ideal 06B10,11A99
Онлайн доступ:	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/760
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Algebra and Discrete Mathematics

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Algebra and Discrete Mathematics

id	oai:ojs.admjournal.luguniv.edu.ua:article-760
record_format	ojs
spelling	oai:ojs.admjournal.luguniv.edu.ua: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
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	2024-04-12T06:26:52Z
last_indexed	2024-04-12T06:26:52Z
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Ideals in \((\mathcal{Z}^{+},\leq_{D})\)

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