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 $leq _C$ on Z+ by ‘ m $leq _C$ 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. 2000 Mathematics Subject Classifications: 06B99,11A99.