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Distributive and Anti-distributive Mendelsohn Triple Systems

Published online by Cambridge University Press:  20 November 2018

Diane M. Donovan ,
Terry S. Griggs ,
Thomas A. McCourt ,
Jakub Opršal  and
David Stanovský
Diane M. Donovan
Affiliation:
Centre for Discrete Mathematics and Computing, University of Queensland, St Lucia 4072, Australia e-mail: dmd@maths.uq.edu.au
Terry S. Griggs
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom e-mail: t.s.griggs@open.ac.uk
Thomas A. McCourt
Affiliation:
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 ITW, United Kingdom e-mail: thomas.mccourt@plymouth.ac.uk
Jakub Opršal
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic e-mail: oprsal@karlin.mff.cuni.cz e-mail: stanovsk@karlin.mff.cuni.cz
David Stanovský
Affiliation:
Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic e-mail: oprsal@karlin.mff.cuni.cz e-mail: stanovsk@karlin.mff.cuni.cz
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Abstract

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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 asmany combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.

Keywords

20N0505B07Mendelsohn triple systemquasigroupdistributiveMoufang loopLoeschian numbers
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 1 , 01 March 2016 , pp. 36 - 49
Copyright
Copyright © Canadian Mathematical Society 2016

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