Algebra universalis

, 80:9 | Cite as

Taylor term does not imply any nontrivial linear one-equality Maltsev condition

  • Alexandr KazdaEmail author


It is known that any finite idempotent algebra that satisfies a nontrivial Maltsev condition must satisfy the linear one-equality Maltsev condition (a variant of the term discovered by Siggers and refined by Kearnes, Marković, and McKenzie):
$$\begin{aligned} t(r,a,r,e)\approx t(a,r,e,a). \end{aligned}$$
We show that if we drop the finiteness assumption, the k-ary weak near unanimity equations imply only trivial linear one-equality Maltsev conditions for every \(k\ge 3\). From this it follows that there is no nontrivial linear one-equality condition that would hold in all idempotent algebras having Taylor terms. Miroslav Olšák has recently shown that there is a weakest nontrivial strong Maltsev condition for idempotent algebras. Olšák has found several such (mutually equivalent) conditions consisting of two or more equations. Our result shows that Olšák’s equation systems cannot be compressed into just one equation.


Maltsev conditions Equational theory Weak near unanimity Siggers term 

Mathematics Subject Classification

08B05 03C05 



  1. 1.
    Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge University Press, New York (1999)zbMATHGoogle Scholar
  2. 2.
    Barto, L., Kozik, M.: Absorbing subalgebras, cyclic terms, and the constraint satisfaction problem. Logical Methods in Computer Science 8(1) (2012).
  3. 3.
    Bergman, C.: Universal algebra: fundamentals and selected topics, 1st edn. Chapman & Hall/CRC Press, Boca Raton (2011)CrossRefGoogle Scholar
  4. 4.
    Kearnes, K., Marković, P., McKenzie, R.: Optimal strong Mal’cev conditions for omitting type 1 in locally finite varieties. Algebra Univers. 72(1), 91–100 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Maróti, M., McKenzie, R.: Existence theorems for weakly symmetric operations. Algebra Univers. 59, 463–489 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Olšák, M.: The weakest nontrivial idempotent equations. Bull. Lond. Math. Soc. 49(6), 1028–1047 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Siggers, M.H.: A strong Mal’cev condition for locally finite varieties omitting the unary type. Algebra Univers. 64(1), 15–20 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Taylor, W.: Varieties obeying homotopy laws. Can. J. Math. 29, 498–527 (1977)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of AlgebraCharles UniversityPragueCzech Republic

Personalised recommendations