Characterization of regular convolutions

Abstract

A convolution is a mapping \(\mathcal{C}\) of the set \(Z^{+}\) of positive integers into the set \({\mathscr{P}}(Z^{+})\) of all subsets of \(Z^{+}\) such that, for any \(n\in Z^{+}\), each member of \(\mathcal{C}(n)\) is a divisor of \(n\). If \(\mathcal{D}(n)\) is the set of all divisors of \(n\), for any \(n\), then \(\mathcal{D}\) is called the Dirichlet's convolution [2]. If \(\mathcal{U}(n)\) is the set of all Unitary(square free) divisors of \(n\), for any \(n\), then \(\mathcal{U}\) is called unitary(square free) convolution. Corresponding to any general convolution \(\mathcal{C}\), we can define a binary relation \(\leq_{\mathcal{C}}\) on \(Z^{+}\) by `\(m\leq_{\mathcal{C}}n\) if and only if \( m\in \mathcal{C}(n)\)'. In this paper, we present a characterization of regular convolution.