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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

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

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
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}