Advertisement

Computer Search for Large Sets of Idempotent Quasigroups

  • Feifei Ma
  • Jian Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5081)

Abstract

A collection of n − 2 idempotent quasigroups of order n is called a large set if any two of them are disjoint, denoted by LIQ(n). While the existence of ordinary LIQ(n) has been extensively studied, the spectrums of large sets of idempotent quasigroups with various identities remain open, for example, large set of Steiner pentagon quasigroups of order 11 which is denoted by LSPQ(11). This paper describes some computer searching efforts seeking to solve such problems. A series of results are obtained, including the non-existence of LSPQ(11).

Keywords

Constraint Satisfaction Problem Combinatorial Design Order Formula Hard Combinatorial Problem Idempotent Quasigroups 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chang, Y.: The Spectrum for Large Sets of Idempotent Quasigroups. Journal of Combinatorial Designs (2000)Google Scholar
  2. 2.
    Colbourn, C.J., Dinitz, J.H.: Handbook of Combinatorial Designs, 2nd edn. CRC Press, Boca Raton (2007)zbMATHGoogle Scholar
  3. 3.
    Dubois, O., Dequen, G.: The Non-existence of (3,1,2)-Conjugate Orthogonal Idempotent Latin Square of Order 10. In: Proc. of the 7th International Conference on Principles and Practice of Constraint Programming (2001)Google Scholar
  4. 4.
    Fujita, M., Slaney, J.K., Bennett, F.: Automatic Generation of Some Results in Finite Algebra. In: Proc. IJCAI, pp. 52–59 (1993)Google Scholar
  5. 5.
    Jia, X., Zhang, J.: A Powerful Technique to Eliminate Isomorphism in Finite Model Search. In: Proc. IJCAR, pp. 318–331 (2006)Google Scholar
  6. 6.
    Lindner, C.C., Stinson, D.R.: Steiner Pentagon Systems. Discrete Math. 52, 67–74 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    McCune, W.: Mace4 Reference Manual and Guide. Technical Memorandum 264, Argonne National Laboratory, Argonne, IL, USA (2003)Google Scholar
  8. 8.
    Slaney, J., Fujita, M., Stickel, M.: Automated Reasoning and Exhaustive Search: Quasigroup Existence Problems. Computers and Mathematics with Applications 29, 115–132 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Teirlinck, L., Lindner, C.C.: The Construction of Large Sets of Idempotent Quasigroups. Eur. J. of Combin. 9, 83–89 (1988)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Zhang, H.: SATO: An Efficient Propositional Prover. In: Proc. of CADE, pp. 272–275 (1997)Google Scholar
  11. 11.
    Zhang, H., Stickel, M.: Implementing the Davis-Putnam Method. Journal of Automated Reasoning 24(1/2), 277–296 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Zhang, J., Zhang, H.: SEM: a System for Enumerating Models. In: Proc. of International Joint Conference on Artificial Intelligence, pp. 11–18 (1995)Google Scholar
  13. 13.
    Zhu, L.: Large Set Problems for Various Idempotent Quasigroups (July 2006)Google Scholar
  14. 14.
    Zhu, L.: Personal Communication (September 2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Feifei Ma
    • 1
    • 2
  • Jian Zhang
    • 1
  1. 1.State Key Laboratory of Computer Science Institute of SoftwareChinese Academy of Sciences 
  2. 2.Chinese Academy of SciencesGraduate University 

Personalised recommendations