Lattice Structures on Z+ Induced by Convolutions

Authors

  • U.M. Swamy
  • Sagi Sankar

Keywords:

Poset, Lattice, Support, Convolution, Multiplicative, Relatively Prime.

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.

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Published

2011-11-27

Issue

Vol. 4 No. 4: (October 2011)

Section

Algebra

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How to Cite

Lattice Structures on Z+ Induced by Convolutions. (2011). European Journal of Pure and Applied Mathematics, 4(4), 424-434. https://www.ejpam.com/index.php/ejpam/article/view/1373