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: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2010-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2010/284875 |
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| _version_ | 1832554321979375616 |
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| author | Mark Farag Brink van der Merwe |
| author_facet | Mark Farag Brink van der Merwe |
| author_sort | Mark Farag |
| collection | DOAJ |
| 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. |
| format | Article |
| id | doaj-art-d92779dc7086456ca959c5781582c61f |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-d92779dc7086456ca959c5781582c61f2025-02-03T05:51:57ZengWileyInternational 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 |
| spellingShingle | Mark Farag Brink van der Merwe The Near-Ring of Lipschitz Functions on a Metric Space International Journal of Mathematics and Mathematical Sciences |
| title | 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_short | The Near-Ring of Lipschitz Functions on a Metric Space |
| title_sort | near ring of lipschitz functions on a metric space |
| 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 |