Skip to main content
SpringerLink
Log in

Complete classification of isotopically invariant varieties of analytic loops defined by regular identities of length four

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

Using the theory of multidimensional three-webs we give a complete classification of isotopically invariant varieties of analytic loops defined by regular identities of length four.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Suschkewitsch, A. K. “On a Generalization of the Associative Law”, Trans. Amer. Math. Soc. 31, No. 1, 204–214 (1929).

    Article  MathSciNet  MATH  Google Scholar 

  2. Moufang, R. “Zur Struktur von Alternativ Körpern”, Math. Ann. 110, No. 1, 416–430 (1935).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bol, G. “Gewebe und Gruppen”, Math. Ann. 114, No. 1, 414–431 (1937).

    Article  MathSciNet  MATH  Google Scholar 

  4. Murdoch, D. C. “Quasi-Groups Which Satisfy Certain Generalized Associative Laws”, Amer. J.Math. 61, No. 2, 509–522 (1939).

    Article  MathSciNet  MATH  Google Scholar 

  5. Albert, A. A. “Quasigroups”. I, Trans.Amer.Math. Soc. 54, No. 3, 507–519 (1943).

  6. Albert, A. A. “Quasigroups”. II, Trans. Amer.Math. Soc. 55, No. 3, 401–409 (1944).

  7. Bruck, R. H. “Some Results in the Theory of Quasigroups”, Trans. Amer.Math. Soc. 56, No. 2, 141–199 (1944).

    Article  MathSciNet  MATH  Google Scholar 

  8. Bruck, R. H. “Contributions to the Theory of Loops”, Trans.Amer.Math. Soc. 60, No. 2, 245–354 (1946).

    Article  MathSciNet  MATH  Google Scholar 

  9. Maltsev, A. I. “Analytische Loops”, Mat. Sb.N. Ser. 36, No. 3, 569–575 (1955) [in Russian].

    MATH  Google Scholar 

  10. Bruck, R. H. A Survey of Binary Systems (Springer Verlag, 1958).

    Book  MATH  Google Scholar 

  11. Belousov, V. D. Grundlagen der Theorie der Quasigruppen und Loops (Nauka, Moscow, 1967) [in Russian].

    MATH  Google Scholar 

  12. Pflugfelder, H. O. Quasigroups and Loops. Introduction, Sigma series in pure mathematics 7 (Heldeman Verlag, Berlin, 1990).

    MATH  Google Scholar 

  13. Chein, O., Pflugfelder, H. O., and Smith, J. D. H. Quasigroups and Loops. Theory and Applications, Sigma series in pure mathematics 8 (Heldeman Verlag, Berlin, 1990).

    MATH  Google Scholar 

  14. Sade, A. “Entropie Demosienne de Multigroupoides et de Quasigroupes”, Ann. Soc. Sci. Bruxelles, Sér. I 73, No. 3, 302–309 (1959).

    MathSciNet  MATH  Google Scholar 

  15. Belousov, V. D. “Balanced Identities in Quasigroups”, Mat. Sb.,N. Ser. 70, 55–97 (1966).

    MathSciNet  MATH  Google Scholar 

  16. Belousov, V. D. “Balanced Identities in Algebras of Quasigroups”, Aequationes Math. 8, No. 1–2, 1–73 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  17. Falcone, E. “Isotopy Invariants in Quasigroups”, Trans.Amer.Math. Soc. 151, No. 2, 511–526 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  18. Vechtomov, V. E. “Über Figuren der Abschließung für eine Klasse universeller Identitäten”, Mat. Issled. 10, No. 2, 36–63 (1975) [in Russian].

    MathSciNet  MATH  Google Scholar 

  19. Akivis, M. A. and Shelekhov, A. M. Geometry and Algebra of Multidimensional Three-webs (Kluwer Academic Publishers, Dordrecht–Boston–London, 1992).

    Book  MATH  Google Scholar 

  20. Akivis, M. A. and Shelekhov, A. M. Multidimensional Three-webs and Their Applications (Tver State University, Tver, 2010) [in Russian].

    MATH  Google Scholar 

  21. Shelekhov, A. M., Lazareva, V. B., and Utkin, A. A. Curvilinear Three-Webs (Tver State University, Tver, 2013) [in Russian].

    MATH  Google Scholar 

  22. Fenyves, F. “Extra Loops”. I, Publ. Math. Debrecen 15, 235–238 (1968).

  23. Fenyves, F. “Extra Loops”. II. On Loops with Identitites of Bol–Moufang Type, Publ.Math. Debrecen 16, 187–192 (1969).

  24. Phillips, J. D. and Vojtechovsky, P. “The Varieities of Loops of Bol–Moufang Type”, Algebra Universalis 54, No. 3, 259–271 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  25. Phillips, J. D. and Vojtechovsky, P. “The Varieities of Quasigroups of Bol–Moufang Type: An Equational Reasoning Approach”, J. Algebra 293, No. 1, 17–33 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  26. Cheban, A. M. “Loops with Identities of Length 4 and Rank 3”. II, in General Algebra and Discrete Geometry (Shtiintsa, Kishinev, 1980), pp. 117–120.

    Google Scholar 

  27. Coté, B., Harvill, B., Huhn, M., and Kirshman, A. “Classification of Loops of Generalized Bol–Moufang Type”, Quasigroup and Related Systems 19, No. 2, 193–206 (2011).

    MathSciNet  MATH  Google Scholar 

  28. Vechtomov, V. E. “On a Type of Universal Identities in Loops”, Abstracts of All-Union Geometrical Symposium in Gomel, 1975, Part II (Gomel, 1975), pp. 256–258.

    Google Scholar 

  29. Shelekhov, A.M. “On Three-Webs with Elastic Coordinate Loops”, Available from VINITI, No. 8465-87 (Kalinin, 1987).

    Google Scholar 

  30. Shelekhov, A. M. “Analytic Solutions of the Functional Equation x(yx) = (xy)x”, Math. Notes 50, No. 4, 1073–1078 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  31. Shelekhov, A. M. “New Closure Conditions and Some Problems in Loop Theory”, Aequationes Math. 41, No. 1, 79–84 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  32. Shelekhov, A. M. “The Isotopically Invariant Loop Variety Lying Between Moufang Loop Variety and Bol Loop Variety”, in Proceedings of the 3rd International Congress in Geometry (Thessaloniki, Greece 1991), pp. 376–384 (1992).

    Google Scholar 

  33. Shelekhov, A. M. “Identities with One Variable in Loops Equivalent to Monoassociativity”, in Problems of Theory of Webs and Quasigroups (Kalinin State Univ., Kalinin, 1985), pp. 89–93.

    Google Scholar 

  34. Shelekhov, A. M. “Classsification of Multidimensional Three-webs by Closure Conditions”, J. of Soviet Mathematics 55, No. 6, 2140–2168 (1991).

    Article  MATH  Google Scholar 

  35. Billig, V. A. and Shelekhov, A. M. “On Classification of Identities withOneVariable in a Smooth Local Loop”, in Webs and Quasigroups (Kalinin State Univ., Kalinin, 1987), pp. 24–32.

    Google Scholar 

  36. Billig, V. A. and Shelekhov, A. M. “Classification of Identities of Length 12 and Order 4 with One Variable in a Local Analytic Loop”, in Webs and Quasigroups (Kalinin State Univ., Kalinin, 1990), pp. 10–18.

    Google Scholar 

  37. Aczel, J. “Quasigroups, Nets and Nomograms”, Adv. Math. 1, No. 3, 383–450 (1965).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tver State University, ul. Zhelyabova 33, Tver, 170100, Russia

    A. M. Shelekhov

Authors
  1. A. M. Shelekhov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. Shelekhov.

Additional information

Original Russian Text © A.M. Shelekhov, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 3, pp. 68–77.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shelekhov, A.M. Complete classification of isotopically invariant varieties of analytic loops defined by regular identities of length four. Russ Math. 61, 58–66 (2017). https://doi.org/10.3103/S1066369X17030070

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X17030070

Keywords

Search

Navigation