Partial orders induced by convolutions

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 for any $nin mathcal{Z^{+}}$, each member of $mathcal{C}(n)$ is a divisor of $n$. If $D(n)$ is the set of all divisors of $n$, for any $n$, then $D$ is called the Dirichlet's convolution. If $U(n)$ is the set of all unitary(square free) divisors of $n$, for any $n$, then $U$ is called unitary(square free) convolution. Corresponding to any general convolution $mathcal{C}$, we can define a binary relation $leq_{mathcal{C}}$ on $mathcal{Z^{+}}$ by ` $mleq_{mathcal{C}}n $ if and only if $ min mathcal{C}(n)$ '. In this paper, we characterize convolutions $mathcal{C}$ for which $leq_{mathcal{C}}$ is a partial order on $mathcal{Z^{+}}$ and discuss the various properties of the partial ordered set $(mathcal{Z^{+}},leq_{mathcal{C}})$ in terms of the convolution $mathcal{C}$.