Abstract
Aichinger et al. (2011) have proved that every finite algebra with a cube-term (equivalently, with a parallelogram-term; equivalently, having few subpowers) is finitely related. Thus finite algebras with cube terms are inherently finitely related—every expansion of the algebra by adding more operations is finitely related. In this paper, we show that conversely, if A is a finite idempotent algebra and every idempotent expansion of A is finitely related, then A has a cube-term. We present further characterizations of the class of finite idempotent algebras having cube-terms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomial-time algorithm for determining if the algebra has a cube-term. We also determine the maximal non-finitely related idempotent clones over A. The number of these clones is finite.
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Aichinger, E., McKenzie, R., Mayr, P.: Finite algebras with cube-terms are finitely related. European J. Math. (2011, in press)
Barto, L.: Finitely related algebras in congruence distributive varieties have near unanimity terms. http://www.karlin.mff.cuni.cz/∼barto/Articles/ZadoriConjecture.pdf (2011). Accessed 13 Sept 2011
Barto, L., Kozik, M.: New conditions for Taylor varieties and CSP. In: Proceedings of 25th IEEE Symposium on Logic in Computer Science, LICS’10, pp. 100–109 (2010)
Berman, J., Idziak, P., Marković, P., McKenzie, R., Valeriote, M., Willard, R.: Varieties with few subalgebras of powers. Trans. Amer. Math. Soc. 362, 1145–1173 (2009)
Davey, B.A., Jackson, M., Pitkethly, J.G., Szabó, Cs.: Finite degree: algebras in general and semigroups in particular. In: Semigroup Forum (2011, in press)
Freese, R., McKenzie, R.: Commutator theory for congruence modular varieties. In: London Mathematical Society Lecture Note Series No. 125. Cambridge University Press, Cambridge (1987)
Horowitz, J.: Results on the computational complexity of linear idempotent Mal’cev conditions. Ph.D. dissertation. McMaster University (2011)
Idziak, P., Marković, P., McKenzie, R., Valeriote, M., Willard, R.: Tractability and learnability arising from algebras with few subpowers. In: Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science, pp. 221–230
Marković, P., McKenzie, R.: Few subpowers, congruence distributivity, and near-unanimity terms. Algebra Universalis (2011, in press)
Valeriote, M.: A subalgebra intersection property for congruence distributive varieties. Can. J. Math. 61(2), 451–464 (2009)
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The second author was supported by Hungarian National Foundation for Scientific Research (OTKA) grants K77409 and PD75475.
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Marković, P., Maróti, M. & McKenzie, R. Finitely Related Clones and Algebras with Cube Terms. Order 29, 345–359 (2012). https://doi.org/10.1007/s11083-011-9232-2
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DOI: https://doi.org/10.1007/s11083-011-9232-2