Skip to main content
SpringerLink
Log in

Characterizations of several Maltsev conditions

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

Tame congruence theory identifies six Maltsev conditions associated with locally finite varieties omitting certain types of local behaviour. Extending a result of Siggers, we show that of these six Maltsev conditions only two of them are equivalent to strong Maltsev conditions for locally finite varieties. Besides omitting the unary type, the only other of these conditions that is strong is that of omitting the unary and affine types.

We also provide novel presentations of some of the above Maltsev conditions.

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. Barto, L.: The collapse of the bounded width hierarchy. J. Logic Comput., to appear

  2. Barto, L., Kozik, M.: Constraint satisfaction problems of bounded width. In: 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), pp. 595–603. IEEE Computer Soc., Los Alamitos, CA (2009)

  3. Barto, L., Kozik, M.: Absorbing subalgebras, cyclic terms, and the constraint satisfaction problem. Log. Methods Comput. Sci. 8(1), 1:07, 27 (2012)

  4. Barto, L., Kozik, M.: Constraint satisfaction problems solvable by local consistency methods. J. ACM 61(1), 3:1–3:19 (2014)

  5. Barto, L., Kozik, M., Niven, T.: The CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to a conjecture of Bang-Jensen and Hell). SIAM J. Comput. 38(5), 1782–1802 (2008/09)

  6. Barto, L., Kozik, M., Stanovský, D.: Mal’tsev conditions, lack of absorption, and solvability. Algebra Universalis, to appear (2014)

  7. Bergman, C.: Universal Algebra, Fundamentals and Selected Topics. Pure and Applied Mathematics (Boca Raton), vol. 301. CRC Press, Boca Raton (2012)

  8. Bulatov, A.A.: Bounded relational width (2009, preprint)

  9. Bulatov, A.A., Valeriote, M.: Recent results on the algebraic approach to the CSP. In: Creignou et al. [11]., pp. 68–92

  10. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra, Graduate Texts in Mathematics, vol. 78. Springer, New York (1981)

  11. Creignou, N., Kolaitis, P.G., Vollmer, H. (eds.): Complexity of Constraints – An Overview of Current Research Themes Result of a Dagstuhl Seminar., Lecture Notes in Computer Science, vol. 5250. Springer (2008)

  12. Freese, R., McKenzie, R.: Maltsev families of varieties closed under join or Maltsev product (2013 preprint)

  13. García, O.C., Taylor, W.: The lattice of interpretability types of varieties. Mem. Amer. Math. Soc. 50 (305) (1984)

  14. Hagemann, J., Mitschke, A.: On n-permutable congruences. Algebra Universalis 3, 8–12 (1973)

  15. Hobby, D., McKenzie, R.: The structure of finite algebras, Contemporary Mathematics vol. 76. American Mathematical Society, Providence (1988). Revised edition (1996)

  16. Jónsson, B.: Algebras whose congruence lattices are distributive. Math. Scand. 21 (1967), 110–121 (1968)

  17. Kearnes, K.A., Kiss, E.W.: The shape of congruence lattices. Mem. Amer. Math. Soc. 222 (1046) (2013)

  18. Kearnes K., Marković P., McKenzie R.: Optimal strong Mal’cev conditions for omitting type 1 in locally finite varieties. Algebra Universalis 72, 91–100 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kearnes K.A., Szendrei Á.: Clones of algebras with parallelogram terms. Internat. J. Algebra Comput. 22(1), 1250005 (2012)

    Article  MathSciNet  Google Scholar 

  20. Mal′cev A.I.: On the general theory of algebraic systems. Mat. Sb. N.S. 35(77), 3–20 (1954)

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. McKenzie, R.N., McNulty, G.F., Taylor, W.F.: Algebras, lattices, varieties. Vol. I. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey (1987)

  23. Nešetřil, J., Siggers, M.: Combinatorial proof that subprojective constraint satisfaction problems are NP-complete. In: Mathematical foundations of computer science 2007, Lecture Notes in Comput. Sci., vol. 4708, pp. 159–170. Springer, Berlin (2007)

  24. Siggers M.H.: A strong Mal’cev condition for locally finite varieties omitting the unary type. Algebra Universalis 64, 15–20 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

    Marcin Kozik

  2. School of Engineering and Computing Sciences, Durham University, Durham, UK

    Andrei Krokhin

  3. Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario, Canada

    Matt Valeriote

  4. Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada

    Ross Willard

Authors
  1. Marcin Kozik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrei Krokhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matt Valeriote
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ross Willard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matt Valeriote.

Additional information

Presented by R. Freese.

The first author was supported by National Science Centre DEC-2011/01/B/ST6/01006. The third and fourth authors were supported by the Natural Sciences and Engineering Research Council of Canada.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kozik, M., Krokhin, A., Valeriote, M. et al. Characterizations of several Maltsev conditions. Algebra Univers. 73, 205–224 (2015). https://doi.org/10.1007/s00012-015-0327-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-015-0327-2

2010 Mathematics Subject Classification

Key words and phrases

Search

Navigation