Abstract
An n-ary operation f : A n → A is called cyclic if it is idempotent and \({f(a_1, a_2, a_3, \ldots , a_n) = f(a_2, a_3, \ldots , a_n, a_1)}\) for every \({a_1, \ldots, a_n \in A}\). We prove that every finite algebra A in a congruence modular variety has a p-ary cyclic term operation for any prime p greater than |A|.
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., 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. Comp. 38, 1782–1802 (2009)
Brauer A., Shockley J.E.: On a problem of Frobenius. J. Reine. Angew. Math. 211, 215–220 (1962)
Bulatov, A.A., Krokhin, A., Jeavons, P.G.: Constraint satisfaction problems and finite algebras. In: Proc. of the 27th Int. Colloquium on Automata, Languages and Programming (ICALP). Lecture Notes in Computer Science, vol. 1853, pp. 272–282. Springer (2000)
Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Graduate Texts in Mathematics, vol. 78. Springer, New York (1981)
Carvalho C., Dalmau V., Marković P., Maróti M.: CD(4) has bounded width. Algebra Universalis 60, 293–307 (2009)
Feder T., Vardi M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory. SIAM J. Computing 28, 57–104 (1998)
Freese, R., McKenzie, R.: Commutator Theory for Congruence Modular Varieties. London Mathematical Society Lecture Note Series, vol. 125. Cambridge University Press, Cambridge (1987)
Gumm H.P.: Geometrical methods in congruence modular varieties. Mem. Amer. Math. Soc. 45, 286 (1983)
Hobby D., McKenzie R.: The Structure of Finite Algebras. Contemporary Mathematics, vol. 76. American Mathematical Society, Providence (1988)
Kiss, E.W., Valeriote, M.: On tractability and congruence distributivity. Logical Methods in Computer Science 3 (2007)
Maltsev A.I.: On the general theory of algebraic systems. Mat. Sb. N.S. 35(77), 3–20 (1954) (Russian)
Maltsev A.I.: On the general theory of algebraic systems. Amer. Math. Soc. Transl. (2) 27, 125–142 (1963) (English translation of Maltsev, 1954)
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. 1. Wadsworth & Brooks/Cole, Monterey (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by E. Kiss.
The first author was supported by the Grant Agency of the Czech Republic under grant no. 201/06/0664 and by the project of Ministry of Education under grant no. MSM 0021620839. The second and fifth authors were supported by the Eduard Čech Center grant LC505. The third author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant nos. T 48809 and K 60148. The fourth author was supported by the US National Science Foundation, grant no. DMS 0604065.
Rights and permissions
About this article
Cite this article
Barto, L., Kozik, M., Maróti, M. et al. Congruence modularity implies cyclic terms for finite algebras. Algebra Univers. 61, 365 (2009). https://doi.org/10.1007/s00012-009-0025-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-009-0025-z