Geodesic
The centre of a Steiner loop and the maxi-Pasch problem
Commentationes Mathematicae Universitatis Carolinae, Tome 61 (2020) no. 4, pp. 535-545.

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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.
DOI : 10.14712/1213-7243.2020.035
Classification : 05B07, 20N05
Keywords: Steiner loop; centre; nucleus; Steiner triple system; Pasch configuration; quadrilateral 
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Kozlik, Andrew R. The centre of a Steiner loop and the maxi-Pasch problem. Commentationes Mathematicae Universitatis Carolinae, Tome 61 (2020) no. 4, pp. 535-545. doi : 10.14712/1213-7243.2020.035. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2020.035/

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