Characterization of regular convolutions

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.