Abstract
We show that the class of locally finite varieties omitting type 1 has the following properties. This class is:
-
(1)
definable by an idempotent, linear, strong Mal’cev condition in a language with one 4-ary function symbol;
-
(2)
not definable by an idempotent, linear, strong Mal’cev condition in a language with only one function symbol of arity strictly less than 4;
-
(3)
definable by an idempotent, linear, strong Mal’cev condition in a language with two 3-ary function symbols;
-
(4)
not definable by an idempotent, linear, strong Mal’cev condition in a language with function symbols of arity less than 4 unless at least two of the symbols have arity 3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barto, L., Kozik, M.: Absorbing subalgebras, cyclic terms, and the constraint satisfaction problem. Log. Methods Comput. Sci. 8 1:07, 27 pp. (2012)
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, 1782–1802 (2008)
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)
Burris, S., Sankappanavar, H.P.: A course in universal algebra. Graduate Texts in Mathematics, vol. 78. Springer, New York (1981)
Hell P., Nešetřil J.: On the complexity of H-colouring. J. Combin. Theory B 48, 92–100 (1990)
Hell P., Nešetřil J., Zhu X.: Duality and polynomial testing of tree homomorphisms. Trans. Amer. Math. Soc. 348, 1281–1297 (1996)
Hobby, D., McKenzie, R.: The structure of finite algebras. Contemporary Mathematics, vol. 76. American Mathematical Society, Providence (1988)
Jovanović J.: On terms describing omitting unary and affine types. Filomat 27, 183–199 (2013)
Kozik, M., Krokhin, A., Valeriote, M. Willard, R.: Characterizations of several Maltsev conditions. (2013, preprint) http://www.math.uwaterloo.ca/~rdwillar/documents/Publications/MaltsevPaper.pdf
Maróti, M. (unpublished) see the lecture of Valeriote, M.: Maltsev conditions for omitting types. presented at the International Conference on Algebras and Lattices, Charles University, Prague, 21.–25. June 2010. http://www.karlin.mff.cuni.cz/~ical/presentations/Valeriote.pdf, page 17
Maróti M., McKenzie R.: Existence theorems for weakly symmetric operations. Algebra Universalis 59, 463–489 (2008)
McKenzie, R., McNulty, G., Taylor, W.: Algebras, lattices, varieties. Vol. I. The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey (1987)
Siggers M.: A strong Mal’cev condition for locally finite varieties omitting the unary type. Algebra Universalis 64, 15–20 (2010)
Taylor W.: Varieties obeying homotopy laws. Canad. J. Math. 29, 498–527 (1977)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by R. Freese.
The second author was supported by the grant no. 174018 of the Ministry of Education and Science of Serbia.
Rights and permissions
About this article
Cite this article
Kearnes, K., Marković, P. & McKenzie, R. Optimal strong Mal’cev conditions for omitting type 1 in locally finite varieties. Algebra Univers. 72, 91–100 (2014). https://doi.org/10.1007/s00012-014-0289-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-014-0289-9