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Keywords:
Steiner loop; centre; nucleus; Steiner triple system; Pasch configuration; quadrilateral
Steiner loop; centre; nucleus; Steiner triple system; Pasch configuration; quadrilateral
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.
References:
[1] Bruck R. H.: A Survey of Binary Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, 20, Reihe: Gruppentheorie, Springer, Berlin, 1971. MR 0093552 | Zbl 0141.01401
[2] Colbourn C. J.: The configuration polytope of $ l$-line configurations in Steiner triple systems. Math. Slovaca 59 (2009), no. 1, 77–108. DOI 10.2478/s12175-008-0111-2 | MR 2471691
[3] Colbourn C. J., Rosa A.: Triple Systems. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. MR 1843379 | Zbl 1030.05017
[4] Danziger P., Mendelsohn E., Grannell M. J., Griggs T. S.: Five-line configurations in Steiner triple systems. Utilitas Math. 49 (1996), 153–159. MR 1396296
[5] Di Paola J. W.: When is a totally symmetric loop a group?. Amer. Math. Monthly 76 (1969), 249–252. DOI 10.1080/00029890.1969.12000185 | MR 0240231
[6] Donovan D.: The centre of a sloop. Combinatorial Mathematics and Combinatorial Computing, Australas. J. Combin. 1 (1990), 83–89. MR 1126969
[7] Donovan D., Rahilly A.: The central spectrum of the order of a Steiner loop. Southeast Asian Bull. Math. 16 (1992), no. 2, 115–121. MR 1205545
[8] Drápal A.: On quasigroups rich in associative triples. Discrete Math. 44 (1983), no. 3, 251–265. DOI 10.1016/0012-365X(83)90189-9 | MR 0696286
[9] Grannell M. J., Griggs T. S., Mendelsohn E.: A small basis for four‐line configurations in Steiner triple systems. J. Combin. Des. 3 (1995), no. 1, 51–59. DOI 10.1002/jcd.3180030107 | MR 1305447
[10] Grannell M. J., Griggs T. S., Whitehead C. A.: The resolution of the anti-Pasch conjecture. J. Combin. Des. 8 (2000), no. 4, 300–309. DOI 10.1002/1520-6610(2000)8:4<300::AID-JCD7>3.0.CO;2-R | MR 1762019
[11] Grannell M. J., Lovegrove G. J.: Maximizing the number of Pasch configurations in a Steiner triple system. Bull. Inst. Combin. Appl. 69 (2013), 23–35. MR 3155869
[12] Gray B. D., Ramsay C.: On the number of Pasch configurations in a Steiner triple system. Bull. Inst. Combin. Appl. 24 (1998), 105–112. MR 1641476
[13] Kaski P., Östergård P. R. J.: The Steiner triple systems of order 19. Math. Comp. 73 (2004), no. 248, 2075–2092. DOI 10.1090/S0025-5718-04-01626-6 | MR 2059752
[14] Kirkman T. P.: On a problem in combinations. Cambridge and Dublin Math. J. 2 (1847), 191–204.
[15] Ling A. C. H., Colbourn C. J., Grannell M. J., Griggs T. S.: Construction techniques for anti-Pasch Steiner triple systems. J. London Math. Soc. (2) 61 (2000), no. 3, 641–657. DOI 10.1112/S0024610700008838 | MR 1765934
[16] McCune W.: Mace4 Reference Manual and Guide. Tech. Memo ANL/MCS-TM-264, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, 2003.
[17] Stinson D. R., Wei Y. J.: Some results on quadrilaterals in Steiner triple systems. Discrete Math. 105 (1992), no. 1–3, 207–219. DOI 10.1016/0012-365X(92)90143-4 | MR 1180204
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