Startseite Mathematik Cyclic and rotational latin hybrid triple systems
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Cyclic and rotational latin hybrid triple systems

  • Andrew Richard Kozlik EMAIL logo
Veröffentlicht/Copyright: 22. September 2017
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 67 Heft 5

Abstract

It is well known that given a Steiner triple system (STS) one can define a binary operation * upon its base set by assigning x * x = x for all x and x * y = z, where z is the third point in the block containing the pair {x, y}. The same can be done for Mendelsohn triple systems (MTS), directed triple systems (DTS) as well as hybrid triple systems (HTS), where (x, y) is considered to be ordered. In the case of STSs and MTSs the operation yields a quasigroup, however this is not necessarily the case for DTSs and HTSs. A DTS or an HTS which induces a quasigroup is said to be Latin. In this paper we study Latin DTSs and Latin HTSs which admit a cyclic or a 1-rotational automorphism. We prove the existence spectra for these systems as well as the existence spectra for their pure variants. As a side result we also obtain the existence spectra of pure cyclic and pure 1-rotational MTSs.

MSC 2010: Primary 05B07; Secondary 20N05
Keywords: directed triple system; hybrid triple system; quasigroup

This work was supported by grant SVV-2013-267317.

(Communicated by Peter Horák)

Acknowledgement

The author would like to thank the reviewers for their comments which helped to improve the paper.

References

[1] Bennett, F. E.—Lindner, C. C.: Quasigroups. In: Handbook of Combinatorial Designs (C. J. Colbourn, J. H. Dinitz, eds.), Chapman & Hall/CRC Press, Boca Raton, 2007, pp. 152–160.Suche in Google Scholar

[2] Bonvicini, S.—Buratti, M.—Rinaldi, G.—Traetta, T.: Some progress on the existence of 1-rotational Steiner triple systems, Des. Codes Cryptogr. 62 (2012), 63–78.10.1007/s10623-011-9491-3Suche in Google Scholar

[3] Buratti, M.: 1-Rotational Steiner triple systems over arbitrary groups, J. Combin. Des. 9 (2001), 215–226.10.1002/jcd.1008Suche in Google Scholar

[4] Cho, C. J.: Rotational Mendelsohn triple systems, Kyungpook Math. J. 26 (1986), 5–9.Suche in Google Scholar

[5] Cho, C. J.—Chae, Y.—Hwang, S. G.: Rotational directed triple systems, J. Korean Math. Soc. 24 (1987), 133–142.Suche in Google Scholar

[6] Cho, C. J.—Han, Y. H.—Kang, S. H.: Cyclic directed triple systems, J. Korean Math. Soc. 23 (1986), 117–125.Suche in Google Scholar

[7] Colbourn, C. J.—Colbourn, M. J.: Disjoint cyclic Mendelsohn triple systems, Ars. Combin. 11 (1981), 3–8.Suche in Google Scholar

[8] Colbourn, C. J.—Jiang, Z.: The spectrum for rotational Steiner triple systems, J. Combin. Des. 4 (1996), 205–217.10.1002/(SICI)1520-6610(1996)4:3<205::AID-JCD4>3.0.CO;2-JSuche in Google Scholar

[9] Colbourn, C. J.—Pulleyblank, W. R.—Rosa, A.: Hybrid triple systems and cubic feedback sets, Graphs. Combin. 5 (1989), 15–28.10.1007/BF01788655Suche in Google Scholar

[10] Drápal, A.—Griggs, T. S.—Kozlik, A. R.: Basics of DTS quasigroups: Algebra, geometry and enumeration, J. Algebra Appl. 14 (2015), 1550089.10.1142/S0219498815500899Suche in Google Scholar

[11] Drápal, A.—Griggs, T. S.—Kozlik, A. R.: Triple systems and binary operations, Discrete Math. 325 (2014), 1–11.10.1016/j.disc.2014.02.007Suche in Google Scholar

[12] Drápal, A.—Griggs, T. S.—Kozlik, A. R.: Pure Latin directed triple systems, Australas. J. Combin. 62 (2015), 59–75.Suche in Google Scholar

[13] Drápal, A.—Kozlik, A.—Griggs, T. S.: Latin directed triple systems, Discrete Math. 312 (2012), 597–607.10.1016/j.disc.2011.04.025Suche in Google Scholar

[14] Gardner, R.—Micale, B.—Pennisi, M.—Zijlstra, R.: Cyclic and rotational hybrid triple systems, Discrete Math. 171 (1997), 121–139.10.1016/S0012-365X(96)00036-2Suche in Google Scholar

[15] Lindner, C. C.—Street, A. P.: Ordered triple systems and transitive quasigroups, Ars. Combin. 17A (1984), 297–306.Suche in Google Scholar

[16] Micale, B.—Pennisi, M.: Cyclic and rotational oriented triple systems, Ars. Combin. 35 (1993), 65–68.Suche in Google Scholar

[17] Mishima, M.: The spectrum of 1-rotational Steiner triple systems over a dicyclic group, Discrete Math. 308 (2008), 2617–2619.10.1016/j.disc.2007.06.001Suche in Google Scholar

[18] Peltesohn, R.: Eine Lösung der beiden Heffterschen Differenzenprobleme, Compos. Math. 6 (1939), 251–257.Suche in Google Scholar

[19] Petersen, J.: Die Theorie der regulären Graphs, Acta. Math. 15 (1891), 193–220.10.1007/BF02392606Suche in Google Scholar

[20] Phelps, K. T.—Rosa, A.: Steiner triple systems with rotational automorphisms, Discrete Math. 33 (1981), 57–66.10.1016/0012-365X(81)90258-2Suche in Google Scholar

[21] Tannenbaum, P.: Abelian Steiner triple systems, Canad. J. Math. 28 (1976), 1251–1268.10.4153/CJM-1976-124-6Suche in Google Scholar

Received: 2015-1-28
Accepted: 2016-5-8
Published Online: 2017-9-22
Published in Print: 2017-10-26

© 2017 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  2. Cyclic and rotational latin hybrid triple systems
  4. Notes on mildly distributive semilattices
  6. On generalized completely distributive posets
  8. Properties of non-associative MV-algebras
  10. On the upper and lower exponential density functions
  12. Quadratic permutations, complete mappings and mutually orthogonal latin squares
  14. On F-groups with the central factor of order p4
  16. Comparison of some families of real functions in porosity terms
  18. Negative interest rates: why and how?
  20. Homoclinic and heteroclinic motions in hybrid systems with impacts
  22. Some fixed point theorems in Branciari metric spaces
  24. S-essential spectra and measure of noncompactness
  26. Unified approach to graphs and metric spaces
  28. On structural properties of porouscontinuous functions
  30. A class of topological spaces between the classes of regular and urysohn spaces
  32. On The betti numbers of oriented Grassmannians and independent semi-invariants of binary forms
  34. Generalized Baskakov type operators
https://doi.org/10.1515/ms-2017-0032
Schlagwörter für diesen Artikel
directed triple system; hybrid triple system; quasigroup

Artikel in diesem Heft

  2. Cyclic and rotational latin hybrid triple systems
  4. Notes on mildly distributive semilattices
  6. On generalized completely distributive posets
  8. Properties of non-associative MV-algebras
  10. On the upper and lower exponential density functions
  12. Quadratic permutations, complete mappings and mutually orthogonal latin squares
  14. On F-groups with the central factor of order p4
  16. Comparison of some families of real functions in porosity terms
  18. Negative interest rates: why and how?
  20. Homoclinic and heteroclinic motions in hybrid systems with impacts
  22. Some fixed point theorems in Branciari metric spaces
  24. S-essential spectra and measure of noncompactness
  26. Unified approach to graphs and metric spaces
  28. On structural properties of porouscontinuous functions
  30. A class of topological spaces between the classes of regular and urysohn spaces
  32. On The betti numbers of oriented Grassmannians and independent semi-invariants of binary forms
  34. Generalized Baskakov type operators
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 16.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0032/pdf?lang=de
Button zum nach oben scrollen