Loops with Abelian Inner Mapping Groups: An Application of Automated Deduction

  • Michael Kinyon
  • Robert Veroff
  • Petr Vojtěchovský
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7788)

Abstract

We describe a large-scale project in applied automated deduction concerned with the following problem of considerable interest in loop theory: If Q is a loop with commuting inner mappings, does it follow that Q modulo its center is a group and Q modulo its nucleus is an abelian group? This problem has been answered affirmatively in several varieties of loops. The solution usually involves sophisticated techniques of automated deduction, and the resulting derivations are very long, often with no higher-level human proofs available.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Belousov, V.D.: Foundations of the Theory of Quasigroups and Loops. Izdat. Nauka, Moscow (1967) (in Russian)Google Scholar
  2. 2.
    Bruck, R.H.: Contributions to the theory of loops. Trans. Amer. Math. Soc. 60, 245–354 (1946)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bruck, R.H.: A Survey of Binary Systems. Springer, Heidelberg (1971)CrossRefMATHGoogle Scholar
  4. 4.
    Csörgő, P.: Abelian inner mappings and nilpotency class greater than two. European J. Combin. 28, 858–867 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Csörgő, P., Drápal, A.: Left conjugacy closed loops of nilpotency class two. Results Math. 47, 242–265 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Csörgő, P., Kepka, T.: On loops whose inner permutations commute. Comment. Math. Univ. Carolin. 45, 213–221 (2004)MathSciNetMATHGoogle Scholar
  7. 7.
    Drápal, A., Kinyon, M.K.: Buchsteiner loops: associators and constructions. arXiv/0812.0412Google Scholar
  8. 8.
    Drápal, A., Vojtěchovský, P.: Explicit constructions of loops with commuting inner mappings. European J. Combin. 29, 1662–1681 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Evans, T.: Homomorphisms of non-associative systems. J. London Math. Soc. 24, 254–260 (1949)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fenyves, F.: Extra loops II. On loops with identities of Bol-Moufang type. Publ. Math. Debrecen 16, 187–192 (1969)MathSciNetMATHGoogle Scholar
  11. 11.
    The GAP Group: GAP – Groups, Algorithms, and Programming, Version 4.4.10 (2007), http://www.gap-system.org
  12. 12.
    Hullot, J.-M.: A catalogue of canonical term rewriting systems. Technical Report CSC 113, SRI International (1980)Google Scholar
  13. 13.
    Kepka, T.: On the abelian inner permutation groups of loops. Comm. Algebra 26, 857–861 (1998)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kepka, T., Phillips, J.D.: Connected transversals to subnormal subgroups. Comment. Math. Univ. Carolin. 38, 223–230 (1997)MathSciNetMATHGoogle Scholar
  15. 15.
    Kepka, T., Kinyon, M.K., Phillips, J.D.: The structure of F-quasigroups. J. Algebra 317, 435–461 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kinyon, M.K., Kunen, K., Phillips, J.D.: Every diassociative A-loop is Moufang. Proc. Amer. Math. Soc. 130, 619–624 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kinyon, M.K., Kunen, K., Phillips, J.D.: A generalization of Moufang and Steiner loops. Algebra Universalis 48, 81–101 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kinyon, M.K., Kunen, K., Phillips, J.D.: Diassociativity in conjugacy closed loops. Comm. Algebra 32, 767–786 (2004)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kinyon, M.K., Kunen, K.: The structure of extra loops. Quasigroups and Related Systems 12, 39–60 (2004)MathSciNetMATHGoogle Scholar
  20. 20.
    Kinyon, M.K., Kunen, K.: Power-associative, conjugacy closed loops. J. Algebra 304, 679–711 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kinyon, M.K., Phillips, J.D., Vojtěchovský, P.: C-loops: extensions and constructions. J. Algebra Appl. 6, 1–20 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kinyon, M.K., Phillips, J.D., Vojtěchovský, P.: When is the commutant of a Bol loop a subloop? Trans. Amer. Math. Soc. 360, 2393–2408 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kinyon, M.K., Veroff, R., Vojtěchovský, P.: Loops with abelian inner mapping groups: an application of automated deduction, Web support (2012), http://www.cs.unm.edu/~veroff/AIM_LC/
  24. 24.
    Knuth, D.E., Bendix, P.B.: Simple word problems in universal algebras. In: Leech, J. (ed.) Proceedings of the Conference on Computational Problems in Abstract Algebras, pp. 263–298. Pergamon Press, Oxford (1970)CrossRefGoogle Scholar
  25. 25.
    Kunen, K.: Moufang quasigroups. J. Algebra 183, 231–234 (1996)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kunen, K.: Quasigroups, loops, and associative laws. J. Algebra 185, 194–204 (1996)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kunen, K.: Alternative loop rings. Comm. Algebra 26, 557–564 (1998)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kunen, K.: G-Loops and permutation groups. J. Algebra 220, 694–708 (1999)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kunen, K.: The structure of conjugacy closed loops. Trans. Amer. Math. Soc. 352, 2889–2911 (2000)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    McCune, W.: Otter 3.0 Reference Manual and Guide. Tech. Report ANL-94/6, Argonne National Laboratory, Argonne, IL (1994), http://www.mcs.anl.gov/AR/otter/
  31. 31.
    McCune, W.: Prover9, version 2009-02A, http://www.cs.unm.edu/~mccune/prover9/
  32. 32.
    Nagy, G., Vojtěchovský, P.: LOOPS: Computing with quasigroups and loops in GAP – a GAP package, version 2.0.0 (2008), http://www.math.du.edu/loops
  33. 33.
    Nagy, G.P., Vojtěchovský, P.: Moufang loops with commuting inner mappings. J. Pure Appl. Algebra 213, 2177–2186 (2009)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Niemenmaa, M.: Finite loops with nilpotent inner mapping groups are centrally nilpotent. Bull. Aust. Math. Soc. 79, 109–114 (2009)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Niemenmaa, M., Kepka, T.: On connected transversals to abelian subgroups in finite groups. Bull. London Math. Soc. 24, 343–346 (1992)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Phillips, J.D.: A short basis for the variety of WIP PACC-loops. Quasigroups Related Systems 14, 73–80 (2006)MathSciNetMATHGoogle Scholar
  37. 37.
    Phillips, J.D.: The Moufang laws, global and local. J. Algebra Appl. 8, 477–492 (2009)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Phillips, J.D., Shcherbacov, V.A.: Cheban loops. J. Gen. Lie Theory Appl. 4, Art. ID G100501, 5 p. (2010)Google Scholar
  39. 39.
    Phillips, J.D., Stanovský, D.: Automated theorem proving in quasigroup and loop theory. AI Commun. 23, 267–283 (2010)MathSciNetMATHGoogle Scholar
  40. 40.
    Phillips, J.D., Stanovský, D.: Bruck loops with abelian inner mapping groups. Comm. Alg. (to appear)Google Scholar
  41. 41.
    Phillips, J.D., Vojtěchovský, P.: The varieties of loops of Bol-Moufang type. Algebra Universalis 54, 259–271 (2005)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Phillips, J.D., Vojtěchovský, P.: The varieties of quasigroups of Bol-Moufang type: an equational reasoning approach. J. Algebra 293, 17–33 (2005)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Phillips, J.D., Vojtěchovský, P.: C-loops: an introduction. Publ. Math. Debrecen 68, 115–137 (2006)MathSciNetMATHGoogle Scholar
  44. 44.
    Phillips, J.D., Vojtěchovský, P.: A scoop from groups: equational foundations for loops. Comment. Math. Univ. Carolin. 49, 279–290 (2008)MathSciNetMATHGoogle Scholar
  45. 45.
    Pflugfelder, H.O.: Quasigroups and Loops: Introduction. Sigma Series in Pure Math., vol. 8. Heldermann Verlag, Berlin (1990)MATHGoogle Scholar
  46. 46.
    Rotman, J.J.: An Introduction to the Theory of Groups, 4th edn. Springer, New York (1995)CrossRefMATHGoogle Scholar
  47. 47.
    Veroff, R.: Using hints to increase the effectiveness of an automated reasoning program: case studies. J. Automated Reasoning 16, 223–239 (1996)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Veroff, R.: Solving open questions and other challenge problems using proof sketches. J. Automated Reasoning 27, 157–174 (2001)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Zhang, H., Bonacina, M.P., Hsiang, J.: PSATO: a distributed propositional prover and its application to quasigroup problems. J. Symbolic Computation 21, 543–560 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michael Kinyon
    • 1
  • Robert Veroff
    • 2
  • Petr Vojtěchovský
    • 1
  1. 1.Department of MathematicsUniversity of DenverDenverUSA
  2. 2.Department of Computer ScienceUniversity of New MexicoAlbuquerqueUSA

Personalised recommendations