The Near-Ring of Lipschitz Functions on a Metric Space

The Near-Ring of Lipschitz Functions on a Metric Space

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 funct...

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Main Authors: Mark Farag, Brink van der Merwe
Format: Article
Language:English
Published: Hindawi Limited 2010-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Online Access:http://dx.doi.org/10.1155/2010/284875
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spelling doaj-d92779dc7086456ca959c5781582c61f2020-11-24T21:28:20ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252010-01-01201010.1155/2010/284875284875The Near-Ring of Lipschitz Functions on a Metric SpaceMark Farag0Brink van der Merwe1Department of Mathematics, Fairleigh Dickinson University, 1000 River Rd., Teaneck, NJ 07666, USADepartment of Computer Science, University of Stellenbosch, Stellenbosch 7602, South AfricaThis 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.http://dx.doi.org/10.1155/2010/284875
collection DOAJ
language English
format Article
sources DOAJ
author Mark Farag
Brink van der Merwe
spellingShingle Mark Farag
Brink van der Merwe
The Near-Ring of Lipschitz Functions on a Metric Space
International Journal of Mathematics and Mathematical Sciences
author_facet Mark Farag
Brink van der Merwe
author_sort Mark Farag
title The Near-Ring of Lipschitz Functions on a Metric Space
title_short The Near-Ring of Lipschitz Functions on a Metric Space
title_full The Near-Ring of Lipschitz Functions on a Metric Space
title_fullStr The Near-Ring of Lipschitz Functions on a Metric Space
title_full_unstemmed The Near-Ring of Lipschitz Functions on a Metric Space
title_sort near-ring of lipschitz functions on a metric space
publisher Hindawi Limited
series International Journal of Mathematics and Mathematical Sciences
issn 0161-1712
1687-0425
publishDate 2010-01-01
description 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.
url http://dx.doi.org/10.1155/2010/284875
work_keys_str_mv AT markfarag thenearringoflipschitzfunctionsonametricspace
AT brinkvandermerwe thenearringoflipschitzfunctionsonametricspace
AT markfarag nearringoflipschitzfunctionsonametricspace
AT brinkvandermerwe nearringoflipschitzfunctionsonametricspace
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