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Distributive and Anti-distributive Mendelsohn Triple Systems

  Published:2015-06-06
 Printed: Mar 2016
  • Diane M. Donovan,
    Centre for Discrete Mathematics and Computing, University of Queensland, St Lucia 4072, AUSTRALIA
  • Terry S. Griggs,
    Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UNITED KINGDOM
  • Thomas A. McCourt,
    School of Computing and Mathematics, Plymouth University, Drake Circus, Plymouth PL4 8AA, UNITED KINGDOM and Heilbronn Institute for Mathematical Research, University of Bristol, University Walk, Bristol BS8 1TW, UNITED KINGDOM
  • Jakub Opršal,
    Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, CZECH REPUBLIC
  • David Stanovský,
    Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, CZECH REPUBLIC
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Abstract

We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for as many combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.
Keywords: Mendelsohn triple system, quasigroup, distributive, Moufang loop, Loeschian numbers Mendelsohn triple system, quasigroup, distributive, Moufang loop, Loeschian numbers
MSC Classifications: 20N05, 05B07 show english descriptions Loops, quasigroups [See also 05Bxx]
Triple systems
20N05 - Loops, quasigroups [See also 05Bxx]
05B07 - Triple systems
 

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