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

Sankar Sagi

Algebra and discrete mathematics, Tome 16 (2013) no. 1, pp. 107-115

Résumé

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.

Détail

Keywords: Partial Order, Lattice, Semi Lattice, Ideal.
Mots-clés : Convolution