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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| Published in: | Algebra and Discrete Mathematics |
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| Date: | 2009 |
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| Format: | Article |
| Language: | English |
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Інститут прикладної математики і механіки НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/153380 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | All difference family structures arise from groups / T. Boykett // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 20–31. — Бібліогр.: 17 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862725373563764736 |
|---|---|
| author | Boykett, T. |
| author_facet | Boykett, T. |
| citation_txt | All difference family structures arise from groups / T. Boykett // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 20–31. — Бібліогр.: 17 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| 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.
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| first_indexed | 2025-12-07T18:52:21Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-153380 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T18:52:21Z |
| publishDate | 2009 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Boykett, T. 2019-06-14T03:41:55Z 2019-06-14T03:41:55Z 2009 All difference family structures arise from groups / T. Boykett // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 20–31. — Бібліогр.: 17 назв. — англ. 1726-3255 https://nasplib.isofts.kiev.ua/handle/123456789/153380 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. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics All difference family structures arise from groups Article published earlier |
| spellingShingle | All difference family structures arise from groups Boykett, T. |
| title | 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_short | All difference family structures arise from groups |
| title_sort | all difference family structures arise from groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/153380 |
| work_keys_str_mv | AT boykettt alldifferencefamilystructuresarisefromgroups |