The Near-Ring of Lipschitz Functions on a Metric Space

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.