Irreducible Elements in $(\mathcal{Z}^{+},\leq_{C})$

Abstract

A convolution is a mapping $\mathcal{C}$ of the set $Z^{+}$ of positive integers into the set $\mathcal{P}(Z^{+})$ of all subsets of $Z^{+}$ such that, for any $n\in Z^{+}$ , each member of $C(n)$ 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 $C$, we can define a binary relation $\leq_{C}$ on $Z^{+}$ by ` $m\leq_{C}n $ if and only if $ m\in C(n)$ '. In this paper, we present irreducible elements in $(\mathcal{Z}^{+},\leq_{\mathcal{C}})$ , where $\leq_{\mathcal{C}}$ is the binary relation induced by the convolution $\mathcal{C}$.