Characterization of regular convolutions

Abstract

A convolution is a mapping C of the set Z + of positive integers into the set P(Z +) of all subsets of Z + such that,for any n ∈ 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 [2]. 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 6C on Z+ by ‘m 6C n if and only if m ∈ C(n)’. In this paper, we present a characterization of regular convolution.