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

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Automated Reasoning and Mathematics

Part of the book series: Lecture Notes in Computer Science ((LNAI,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.

Dedicated to the memory of William McCune (1953–2011).

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References

  1. Belousov, V.D.: Foundations of the Theory of Quasigroups and Loops. Izdat. Nauka, Moscow (1967) (in Russian)

    Google Scholar 

  2. Bruck, R.H.: Contributions to the theory of loops. Trans. Amer. Math. Soc. 60, 245–354 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bruck, R.H.: A Survey of Binary Systems. Springer, Heidelberg (1971)

    Book  MATH  Google Scholar 

  4. Csörgő, P.: Abelian inner mappings and nilpotency class greater than two. European J. Combin. 28, 858–867 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Csörgő, P., Drápal, A.: Left conjugacy closed loops of nilpotency class two. Results Math. 47, 242–265 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Csörgő, P., Kepka, T.: On loops whose inner permutations commute. Comment. Math. Univ. Carolin. 45, 213–221 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Drápal, A., Kinyon, M.K.: Buchsteiner loops: associators and constructions. arXiv/0812.0412

    Google Scholar 

  8. Drápal, A., Vojtěchovský, P.: Explicit constructions of loops with commuting inner mappings. European J. Combin. 29, 1662–1681 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Evans, T.: Homomorphisms of non-associative systems. J. London Math. Soc. 24, 254–260 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fenyves, F.: Extra loops II. On loops with identities of Bol-Moufang type. Publ. Math. Debrecen 16, 187–192 (1969)

    MathSciNet  MATH  Google Scholar 

  11. The GAP Group: GAP – Groups, Algorithms, and Programming, Version 4.4.10 (2007), http://www.gap-system.org

  12. Hullot, J.-M.: A catalogue of canonical term rewriting systems. Technical Report CSC 113, SRI International (1980)

    Google Scholar 

  13. Kepka, T.: On the abelian inner permutation groups of loops. Comm. Algebra 26, 857–861 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kepka, T., Phillips, J.D.: Connected transversals to subnormal subgroups. Comment. Math. Univ. Carolin. 38, 223–230 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Kepka, T., Kinyon, M.K., Phillips, J.D.: The structure of F-quasigroups. J. Algebra 317, 435–461 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kinyon, M.K., Kunen, K., Phillips, J.D.: Every diassociative A-loop is Moufang. Proc. Amer. Math. Soc. 130, 619–624 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kinyon, M.K., Kunen, K., Phillips, J.D.: A generalization of Moufang and Steiner loops. Algebra Universalis 48, 81–101 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kinyon, M.K., Kunen, K., Phillips, J.D.: Diassociativity in conjugacy closed loops. Comm. Algebra 32, 767–786 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kinyon, M.K., Kunen, K.: The structure of extra loops. Quasigroups and Related Systems 12, 39–60 (2004)

    MathSciNet  MATH  Google Scholar 

  20. Kinyon, M.K., Kunen, K.: Power-associative, conjugacy closed loops. J. Algebra 304, 679–711 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kinyon, M.K., Phillips, J.D., Vojtěchovský, P.: C-loops: extensions and constructions. J. Algebra Appl. 6, 1–20 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  25. Kunen, K.: Moufang quasigroups. J. Algebra 183, 231–234 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kunen, K.: Quasigroups, loops, and associative laws. J. Algebra 185, 194–204 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kunen, K.: Alternative loop rings. Comm. Algebra 26, 557–564 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kunen, K.: G-Loops and permutation groups. J. Algebra 220, 694–708 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kunen, K.: The structure of conjugacy closed loops. Trans. Amer. Math. Soc. 352, 2889–2911 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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. McCune, W.: Prover9, version 2009-02A, http://www.cs.unm.edu/~mccune/prover9/

  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. Nagy, G.P., Vojtěchovský, P.: Moufang loops with commuting inner mappings. J. Pure Appl. Algebra 213, 2177–2186 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Niemenmaa, M.: Finite loops with nilpotent inner mapping groups are centrally nilpotent. Bull. Aust. Math. Soc. 79, 109–114 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Niemenmaa, M., Kepka, T.: On connected transversals to abelian subgroups in finite groups. Bull. London Math. Soc. 24, 343–346 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  36. Phillips, J.D.: A short basis for the variety of WIP PACC-loops. Quasigroups Related Systems 14, 73–80 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Phillips, J.D.: The Moufang laws, global and local. J. Algebra Appl. 8, 477–492 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Phillips, J.D., Shcherbacov, V.A.: Cheban loops. J. Gen. Lie Theory Appl. 4, Art. ID G100501, 5 p. (2010)

    Google Scholar 

  39. Phillips, J.D., Stanovský, D.: Automated theorem proving in quasigroup and loop theory. AI Commun. 23, 267–283 (2010)

    MathSciNet  MATH  Google Scholar 

  40. Phillips, J.D., Stanovský, D.: Bruck loops with abelian inner mapping groups. Comm. Alg. (to appear)

    Google Scholar 

  41. Phillips, J.D., Vojtěchovský, P.: The varieties of loops of Bol-Moufang type. Algebra Universalis 54, 259–271 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Phillips, J.D., Vojtěchovský, P.: The varieties of quasigroups of Bol-Moufang type: an equational reasoning approach. J. Algebra 293, 17–33 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Phillips, J.D., Vojtěchovský, P.: C-loops: an introduction. Publ. Math. Debrecen 68, 115–137 (2006)

    MathSciNet  MATH  Google Scholar 

  44. Phillips, J.D., Vojtěchovský, P.: A scoop from groups: equational foundations for loops. Comment. Math. Univ. Carolin. 49, 279–290 (2008)

    MathSciNet  MATH  Google Scholar 

  45. Pflugfelder, H.O.: Quasigroups and Loops: Introduction. Sigma Series in Pure Math., vol. 8. Heldermann Verlag, Berlin (1990)

    MATH  Google Scholar 

  46. Rotman, J.J.: An Introduction to the Theory of Groups, 4th edn. Springer, New York (1995)

    Book  MATH  Google Scholar 

  47. Veroff, R.: Using hints to increase the effectiveness of an automated reasoning program: case studies. J. Automated Reasoning 16, 223–239 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  48. Veroff, R.: Solving open questions and other challenge problems using proof sketches. J. Automated Reasoning 27, 157–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Denver, Denver, CO, 80208, USA

    Michael Kinyon & Petr Vojtěchovský

  2. Department of Computer Science, University of New Mexico, Albuquerque, NM, 87131, USA

    Robert Veroff

Authors
  1. Michael Kinyon
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  2. Robert Veroff
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  3. Petr Vojtěchovský
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Verona, 37134, Verona, Italy

    Maria Paola Bonacina

  2. Artificial Intelligence Center, SRI International, 333 Ravenswood Avenue, 94025, Menlo Park, CA, USA

    Mark E. Stickel

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Kinyon, M., Veroff, R., Vojtěchovský, P. (2013). Loops with Abelian Inner Mapping Groups: An Application of Automated Deduction. In: Bonacina, M.P., Stickel, M.E. (eds) Automated Reasoning and Mathematics. Lecture Notes in Computer Science(), vol 7788. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36675-8_8

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  • DOI: https://doi.org/10.1007/978-3-642-36675-8_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36674-1

  • Online ISBN: 978-3-642-36675-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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