Skip to main content
SpringerLink
Log in

Mitschke’s theorem is sharp

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

Mitschke showed that a variety with an m-ary near-unanimity term has Jónsson terms \(t_0, \dots , t _{2m-4} \) witnessing congruence distributivity. We show that Mitschke’s result is sharp. We also evaluate the best possible number of Day terms witnessing congruence modularity. More generally, we characterize exactly the best bounds for many congruence identities satisfied by varieties with an m-ary near-unanimity term. Finally we present some simple observations about terms with just one “dissenter”, a generalization of a minority term.

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. Baker, K.A.: Congruence-distributive polynomial reducts of lattices. Algebra Univ. 9, 142–145 (1979)

    Article  MathSciNet  Google Scholar 

  2. Baker, K.A., Pixley, A.F.: Polynomial interpolation and the Chinese remainder theorem for algebraic systems. Math. Z. 143, 165–174 (1975)

    Article  MathSciNet  Google Scholar 

  3. Barto, L.: Finitely related algebras in congruence distributive varieties have near unanimity terms. Can. J. Math. 65, 3–21 (2013)

    Article  MathSciNet  Google Scholar 

  4. Barto, L., Kozik, M.: Absorption in universal algebra and CSPV. In: The Constraint Satisfaction Problem: Complexity and Approximability, Dagstuhl Follow-Ups, 7 (Schloss Dagstuhl–Leibniz Zentrum für Informatik, Wadern, 2017), 45–77

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

    Article  MathSciNet  Google Scholar 

  6. Campanella, M., Conley, S., Valeriote, M.: Preserving near unanimity terms under products. Algebra Univ. 76, 293–300 (2016)

    Article  MathSciNet  Google Scholar 

  7. Day, A.: A characterization of modularity for congruence lattices of algebras. Can. Math. Bull. 12, 167–173 (1969)

    Article  MathSciNet  Google Scholar 

  8. Freese, R., Valeriote, M.A.: On the complexity of some Maltsev conditions. Int. J. Algebra Comput. 19, 41–77 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Kaarli, K., Pixley, A.F.: Polynomial Completeness in Algebraic Systems. Chapman & Hall/CRC, Boca Raton (2001)

    MATH  Google Scholar 

  11. Kazda, A., Kozik, M., McKenzie, R., Moore, M.: Absorption and directed Jónsson terms. In: Czelakowski, J. (ed.) Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science, Outstanding Contributions to Logic 16, pp. 203–220. Springer, Cham (2018)

  12. Kazda, A., Opršal, J., Valeriote, M., Zhuk, D.: Deciding the existence of minority terms. Can. Math. Bull. 63, 577–591 (2020)

    Article  MathSciNet  Google Scholar 

  13. Lipparini, P.: Relation identities in 3-distributive varieties, Algebra Universalis 80. Paper No. 55, 20 (2019)

  14. Lipparini, P.: The distributivity spectrum of Baker’s variety. J. Aust. Math. Soc. 110, 119–144 (2021)

    Article  MathSciNet  Google Scholar 

  15. Lipparini, P.: Day’s Theorem is sharp for \(n\) even, 1–60 (2021). arXiv:1902.05995v5

  16. Lipparini, P.: Between an \(n\)-ary and an \(n{+}1\)-ary near-unanimity term, 1–15 (2021). arXiv:2105.14041

  17. Lipparini, P.: Mitschke’s Theorem is sharp, 1–21 (2021). arXiv:2105.14041

  18. McKenzie, R.N., McNulty, G.F., Taylor, W. F.: Algebras, Lattices, Varieties. Vol. I (Wadsworth and Brooks/Cole Advanced Books and Software, 1987), corrected reprint with additional bibliography (AMS Chelsea Publishing/American Mathematical Society, 2018)

  19. Mitschke, A.: Implication algebras are \(3\)-permutable and \(3\)-distributive. Algebra Univ. 1, 182–186 (1971)

    Article  MathSciNet  Google Scholar 

  20. Mitschke, A.: Near unanimity identities and congruence distributivity in equational classes. Algebra Univ. 8, 29–32 (1978)

    Article  MathSciNet  Google Scholar 

  21. Olšák, M.: The weakest nontrivial idempotent equations. Bull. Lond. Math. Soc. 49, 1028–1047 (2017)

    Article  MathSciNet  Google Scholar 

  22. Sequeira, L.: Near-unanimity is decomposable. Algebra Univ. 50, 157–164 (2003)

    Article  MathSciNet  Google Scholar 

  23. Tschantz, S.T.: More conditions equivalent to congruence modularity. In: Universal Algebra and Lattice Theory, 270–282, Lecture Notes in Math. 1149 (1985)

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica Viale della Ricerca Quasi Unanime Università di Roma “Tor Vergata”, 00133, Rome, Italy

    Paolo Lipparini

Authors
  1. Paolo Lipparini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paolo Lipparini.

Additional information

Communicated by Presented by L. Barto.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Work performed under the auspices of G.N.S.A.G.A. Work partially supported by PRIN 2012 “Logica, Modelli e Insiemi”. The author acknowledges the MIUR Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lipparini, P. Mitschke’s theorem is sharp. Algebra Univers. 83, 7 (2022). https://doi.org/10.1007/s00012-021-00762-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-021-00762-1

Keywords

Mathematics Subject Classification

Search

Navigation