The centre of a Steiner loop and the maxi-Pasch problem

Andrew R. Kozlik

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 4, page 535-545
  • ISSN: 0010-2628

Abstract

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A binary operation “ · ” which satisfies the identities x · e = x , x · x = e , ( x · y ) · x = y and x · y = y · 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 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.

How to cite

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Kozlik, Andrew R.. "The centre of a Steiner loop and the maxi-Pasch problem." Commentationes Mathematicae Universitatis Carolinae 61.4 (2020): 535-545. <http://eudml.org/doc/297170>.

@article{Kozlik2020,
abstract = {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 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.},
author = {Kozlik, Andrew R.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Steiner loop; centre; nucleus; Steiner triple system; Pasch configuration; quadrilateral},
language = {eng},
number = {4},
pages = {535-545},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The centre of a Steiner loop and the maxi-Pasch problem},
url = {http://eudml.org/doc/297170},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Kozlik, Andrew R.
TI - The centre of a Steiner loop and the maxi-Pasch problem
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 4
SP - 535
EP - 545
AB - 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 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.
LA - eng
KW - Steiner loop; centre; nucleus; Steiner triple system; Pasch configuration; quadrilateral
UR - http://eudml.org/doc/297170
ER -

References

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