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

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.