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The Near-Ring of Lipschitz Functions on a Metric Space

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  • Mark Farag
  • Brink van der Merwe

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.

Suggested Citation

  • Mark Farag & Brink van der Merwe, 2010. "The Near-Ring of Lipschitz Functions on a Metric Space," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2010, pages 1-14, July.
    • Handle: RePEc:hin:jijmms:284875
    DOI: 10.1155/2010/284875
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