All difference family structures arise from groups
Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay's work with planar nearrings to investigate planar nearrings with nonassociative additiv...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-7662018-04-04T08:31:48Z All difference family structures arise from groups Boykett, Tim Algebras, 2-design, difference family, nonassociative Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay's work with planar nearrings to investigate planar nearrings with nonassociative additive structure. Thus we are led to the question of nonassociative difference families.Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalized difference family constructions could be made, using the standard basis of translation and difference.We determine minimal axioms for a difference family structure to give a 2-design. Using these minimal axioms we show that we obtain quasigroups. These quasigroups are shown to be isotopic to groups and the derived 2-designs from the nonassociative difference family are identical to the 2-designs from the isotopic groups. Thus all difference families arise from groups.This result will be of interest to those using nonstandard algebras as bases for defining 2-designs. Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/766 Algebra and Discrete Mathematics; Vol 8, No 1 (2009) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/766/296 Copyright (c) 2018 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
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OJS |
| language |
English |
| topic |
Algebras 2-design difference family nonassociative |
| spellingShingle |
Algebras 2-design difference family nonassociative Boykett, Tim All difference family structures arise from groups |
| topic_facet |
Algebras 2-design difference family nonassociative |
| format |
Article |
| author |
Boykett, Tim |
| author_facet |
Boykett, Tim |
| author_sort |
Boykett, Tim |
| title |
All difference family structures arise from groups |
| title_short |
All difference family structures arise from groups |
| title_full |
All difference family structures arise from groups |
| title_fullStr |
All difference family structures arise from groups |
| title_full_unstemmed |
All difference family structures arise from groups |
| title_sort |
all difference family structures arise from groups |
| description |
Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay's work with planar nearrings to investigate planar nearrings with nonassociative additive structure. Thus we are led to the question of nonassociative difference families.Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalized difference family constructions could be made, using the standard basis of translation and difference.We determine minimal axioms for a difference family structure to give a 2-design. Using these minimal axioms we show that we obtain quasigroups. These quasigroups are shown to be isotopic to groups and the derived 2-designs from the nonassociative difference family are identical to the 2-designs from the isotopic groups. Thus all difference families arise from groups.This result will be of interest to those using nonstandard algebras as bases for defining 2-designs. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/766 |
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AT boyketttim alldifferencefamilystructuresarisefromgroups |
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2024-04-12T06:25:49Z |
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2024-04-12T06:25:49Z |
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