Skip to main content
SpringerLink
Log in

A relational description of higher commutators in Mal’cev varieties

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

We give a relational description of higher commutator operators, which were introduced by Bulatov, in varieties with a Mal’cev term. Furthermore, we use this result to prove that for every algebra with a Mal’cev term, there exists a largest clone containing the Mal’cev operation and having the same congruence lattice and the same higher commutator operators as the original algebra. We also give a local variant of this theorem.

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. Aichinger E., Mayr P., McKenzie R.: On the number of finite algebraic structures. J. Eur. Math. Soc. (JEMS) 16, 1673–1686 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aichinger E., Mudrinski N.: Some applications of higher commutators in Mal’cev algebras. Algebra Universalis 63, 367–403 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aichinger E., Mudrinski N.: On various concepts of nilpotence for expansions of groups. Publ. Math. Debrecen 83, 583–604 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berman J., Idziak P., Marković P., McKenzie R., Valeriote M., Willard R.: Varieties with few subalgebras of powers. Trans. Amer. Math. Soc. 362, 1445–1473 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bulatov, A.: On the number of finite Mal’tsev algebras. In: Contributions to general algebra, 13 (Velké Karlovice, 1999/Dresden, 2000), pp. 41–54. Heyn, Klagenfurt (2001)

  6. Bulatov A.A.: Three-element Mal’tsev algebras. Acta Sci. Math. (Szeged) 71(3-4), 469–500 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Bulatov A.A., Idziak P.M.: Counting Mal’tsev clones on small sets. Discrete Math. 268, 59–80 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Freese, R., McKenzie, R.: Commutator theory for congruence modular varieties, London Mathematical Society Lecture Note Series, vol. 125. Cambridge University Press, Cambridge (1987)

  9. Goldstern M., Pinsker M.: A survey of clones on infinite sets. Algebra Universalis 59, 365–403 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Idziak P.M.: Clones containing Mal’tsev operations. Internat. J. Algebra Comput. 9, 213–226 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mudrinski, N.: On polynomials in Mal’cev algebras. Ph.D. thesis, University of Novi Sad (2009)

  12. Romov B.: Galois correspondence between iterative post algebras and relations on an infinite set. Cybernetics 13, 377–379 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shaw J.: Commutator relations and the clones of finite groups. Algebra Universalis 72, 29–52 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Werner H.: A Mal’cev condition for admissible relations. Algebra Universalis 3, 263 (1973)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Algebra, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Praha, Czech Republic

    Jakub Opršal

  2. Institute for Algebra, Johannes Kepler University, 4040, Linz, Austria

    Jakub Opršal

Authors
  1. Jakub Opršal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jakub Opršal.

Additional information

Presented by K. Kearnes.

Supported by Austrian Science Fund (FWF), P24077: Algebraic approaches to the description of Mal’cev clones; Czech Science Foundation, GAČR 13-01832S; and Charles University in Prague, project SVV-2014-260107.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Opršal, J. A relational description of higher commutators in Mal’cev varieties. Algebra Univers. 76, 367–383 (2016). https://doi.org/10.1007/s00012-016-0391-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-016-0391-2

2010 Mathematics Subject Classification

Key words and phrases

Search

Navigation