International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 284875, 14 pages
doi:10.1155/2010/284875
Research Article

The Near-Ring of Lipschitz Functions on a Metric Space

Mark Farag1 and Brink van der Merwe2

1Department of Mathematics, Fairleigh Dickinson University, 1000 River Rd., Teaneck, NJ 07666, USA
2Department of Computer Science, University of Stellenbosch, Stellenbosch 7602, South Africa

Received 6 August 2009; Revised 3 April 2010; Accepted 25 April 2010

Academic Editor: Francois Goichot

Copyright © 2010 Mark Farag and Brink van der Merwe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper treats near-rings of zero-preserving Lipschitz functions on metric spaces that are also abelian groups, using pointwise addition of functions as addition and composition of functions as multiplication. We identify a condition on the metric ensuring that the set of all such Lipschitz functions is a near-ring, and we investigate the complications that arise from the lack of left distributivity in the resulting right near-ring. We study the behavior of the set of invertible Lipschitz functions, and we initiate an investigation into the ideal structure of normed near-rings of Lipschitz functions. Examples are given to illustrate the results and to demonstrate the limits of the theory.