Ideals in (Z⁺, ≤D)

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 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) '. A general convolution may not induce a lattice on Z⁺. However most of the convolutions induce a meet semi lattice structure on Z⁺. In this paper we consider a general meet semi lattice and study it's ideals and extend these to (Z⁺, ≤D), where D is the Dirichlet's convolution.