Lattice Structures on Z+ Induced by Convolutions

Abstract

A Convolution C is a mapping of the set Z+ of positive integers into the power set P(Z+) such that every member of C(n) is a divisor of n. If for any n, D(n) is the set of all positive divisors of n , then D is called the Dirichlet’s convolution. It is well known that Z+ has the structure of a distributive lattice with respect to the division order. Corresponding to any general convolution C,
one can define a binary relation ≤C on Z+ by ‘ m ≤C n if and only if m ∈ C(n)’ . In this paper we characterize Convolutions C which induce partial orders with respect to which Z+ has the structure of a semi lattice or lattice and various lattice theoretic properties are discussed in terms of convolution.