Skip to main content
SpringerLink
Log in

Mal’tsev conditions, lack of absorption, and solvability

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

We provide a new characterization of several Mal’tsev conditions for locally finite varieties using hereditary term properties. We show a particular example of how a lack of absorption causes collapse in the Mal’tsev hierarchy, and point out a connection between solvability and the lack of absorption. As a consequence, we provide a new and conceptually simple proof of a result of Hobby and McKenzie, saying that locally finite varieties with a Taylor term possess a term which is Mal’tsev on blocks of every solvable congruence in every finite algebra in the variety.

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

Reference

  1. Barto, L., Kozik. M.: Absorbing subalgebras, cyclic terms and the constraint satisfaction problem. Log. Methods Comput. Sci. 8, 1–26 (2012)

  2. Barto L., Kozik M.: Constraint satisfaction problems solvable by local consistency methods. J. ACM 61, 1–19 (2014)

    Article  MathSciNet  Google Scholar 

  3. Bergman, C.: Universal algebra: Fundamentals and Selected Topics. Chapman & Hall/CRC Press (2011)

  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)

  5. Bulatov, A., Jeavons, P.: Algebraic structures in combinatorial problems. Technical Report MATH-AL-4-2001, Technische Universität Dresden (2001)

  6. Bulatov A., Jeavons P., Krokhin A.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 720–742 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Freese, R., McKenzie, R.: Commutator Theory for Congruence Modular Varieties. London Mathematical Society Lecture Note Series 125. Cambridge University Press, Cambridge (1987)

  8. García, O.C., Taylor, W.: The lattice of interpretability types of varieties. Mem. Amer. Math. Soc. 50 (1984)

  9. Hobby, D., McKenzie, R.: The Structure of Finite Algebras. Contemporary Mathematics 76, American Mathematical Society, Providence (1988)

  10. Idziak, P., Marković, P., McKenzie, R., Valeriote, M., Willard, R.: Tractability and learnability arising from algebras with few subpowers. SIAM J. Comput. 39, 3023–3037 (2010)

  11. Kearnes, K., Szendrei, Á.: The relationship between two commutators. Internat. J. Algebra Comput. 8, 497–531 (1998)

  12. Kearnes, K., Szendrei, Á.: Clones of algebras with parallelogram terms. Internat. J. Algebra Comput. 22 (2012)

  13. Kozik, M., Krokhin, A., Valeriote, M., Willard, R.: Characterizations of several Maltsev conditions, Algebra Universalis 73, 205–224 (2015)

  14. Marković, P., Maróti, M., McKenzie, R.: Finitely related clones and algebras with cube terms. Order 29, 345–359 (2012)

  15. Maróti M., McKenzie R.: Existence theorems for weakly symmetric operations. Algebra Universalis 59, 463–489 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. McKenzie, R., Snow, J.: Congruence modular varieties: commutator theory and its uses. In: Structural theory of automata, semigroups, and universal algebra, NATO Sci. Ser. II Math. Phys. Chem. 207, 273–329, Springer, Dordrecht (2005)

  17. Opršal, J., personal communication

  18. Stronkowski, M., Stanovský, D.: Embedding general algebras into modules. Proc. Amer. Math. Soc. 138, 2687–2699 (2010)

  19. Szendrei, Á: Modules in general algebra. Contributions to general algebra 10, Heyn, Klagenfurt, pp. 41–53 (1998)

  20. Taylor, W.: Varieties obeying homotopy laws. Canad. J. Math. 29, 498–527 (1977)

  21. Willard, R.: A finite basis theorem for residually finite, congruence meet-semidistributive varieties. J. Symbolic Logic 65, 187–200 (2000)

  22. Sixty four problems in universal algebra. http://www.math.u-szeged.hu/confer/algebra/2001/progr.html

Download references

Author information

Authors and Affiliations

  1. Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675, Praha 8, Czech Republic

    Libor Barto & David Stanovský

  2. Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Golebia 24, Kraków, Poland

    Marcin Kozik

Authors
  1. Libor Barto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcin Kozik
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Stanovský
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Libor Barto.

Additional information

Presented by G. McNulty.

Dedicated to Brian Davey on the occasion of his 65th birthday

Research partially supported by the GAČR grant 13-01832S (Barto, Stanovský), by the National Science Centre based on DEC-2011/01/B/ST6/01006 (Kozik), and by the Czech-Polish cooperation grant 7AMB13PL013 (Barto, Stanovský) and 8829/R13/R14 (Kozik).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barto, L., Kozik, M. & Stanovský, D. Mal’tsev conditions, lack of absorption, and solvability. Algebra Univers. 74, 185–206 (2015). https://doi.org/10.1007/s00012-015-0338-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-015-0338-z

2010 Mathematics Subject Classification

Key words and phrases

Search

Navigation