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Algebra Discrete Math., 2013, Volume 16, Issue 1, Pages 107–115 (Mi adm439)  

This article is cited in 1 scientific paper (total in 1 paper)

RESEARCH ARTICLE

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

Sankar Sagi

Assistant Professor of Mathematics, College of Applied Sciences, Sohar, Sultanate of Oman

Abstract: 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.

Keywords: Partial Order, Lattice, Semi Lattice, Convolution, Ideal.

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Bibliographic databases:
MSC: 06B10,11A99
Received: 17.12.2011
Revised: 27.03.2013
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Citation: Sankar Sagi, “Ideals in $(\mathcal{Z}^{+},\leq_{D})$”, Algebra Discrete Math., 16:1 (2013), 107–115

Citation in format AMSBIB
\Bibitem{Sag13}
\by Sankar~Sagi
\paper Ideals in $(\mathcal{Z}^{+},\leq_{D})$
\jour Algebra Discrete Math.
\yr 2013
\vol 16
\issue 1
\pages 107--115
\mathnet{http://mi.mathnet.ru/adm439}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3184703}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Sankar Sagi, “Characterization of regular convolutions”, Algebra Discrete Math., 25:1 (2018), 147–156  mathnet
  • Algebra and Discrete Mathematics
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