| Title:
|
The centre of a Steiner loop and the maxi-Pasch problem (English) |
| Author:
|
Kozlik, Andrew R. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
535-545 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A binary operation ``$\cdot$'' which satisfies the identities $x\cdot e = x$, $x \cdot x = e$, $(x \cdot y) \cdot x = y$ and $x \cdot y = y \cdot x$ is called a Steiner loop. This paper revisits the proof of the necessary and sufficient conditions for the existence of a Steiner loop of order $n$ with centre of order $m$ and discusses the connection of this problem to the question of the maximum number of Pasch configurations which can occur in a Steiner triple system (STS) of a given order. An STS which attains this maximum for a given order is said to be {\it maxi-Pasch}. We show that loop factorization preserves the maxi-Pasch property and find that the Steiner loops of all currently known maxi-Pasch Steiner triple systems have centre of maximum possible order. (English) |
| Keyword:
|
Steiner loop |
| Keyword:
|
centre |
| Keyword:
|
nucleus |
| Keyword:
|
Steiner triple system |
| Keyword:
|
Pasch configuration |
| Keyword:
|
quadrilateral |
| MSC:
|
05B07 |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 07332727 |
| idMR:
|
MR4230958 |
| DOI:
|
10.14712/1213-7243.2020.035 |
| . |
| Date available:
|
2021-02-25T12:44:43Z |
| Last updated:
|
2023-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148663 |
| . |
| Reference:
|
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