Partial orders induced by convolutions

Sagi Sankar, U M Swamy


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 $n\in \mathcal{Z^{+}}$, 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 ` $m\leq_{\mathcal{C}}n $ if and only if $ m\in \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 $\mathcal{C}$.

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