Abstract
We study the problem of whether a given finite algebra with finitely many basic operations contains a cube term; we give both structural and algorithmic results. We show that if such an algebra has a cube term then it has a cube term of dimension at most N, where the number N depends on the arities of basic operations of the algebra and the size of the basic set. For finite idempotent algebras we give a tight bound on N that, in the special case of algebras with more than \(\left( {\begin{array}{c}|A|\\ 2\end{array}}\right) \) basic operations, improves an earlier result of K. Kearnes and Á. Szendrei. On the algorithmic side, we show that deciding the existence of cube terms is in P for idempotent algebras and in EXPTIME in general. Since an algebra contains a k-ary near unanimity operation if and only if it contains a k-dimensional cube term and generates a congruence distributive variety, our algorithm also lets us decide whether a given finite algebra has a near unanimity operation.
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Barto, L., Kazda, A.: Deciding absorption. Int. J. Algebra Comput. 26(5), 1033–1060 (2016)
Bergman, C.: Universal algebra: fundamentals and selected topics, 1st edn. Chapman & Hall/CRC Press, Boca Raton and New York and Abingdon (2011)
Berman, J., Idziak, P., Markovic, P., McKenzie, R., Valeriote, M., Willard, R.: Varieties with few subalgebras of powers. Trans. Am. Math. Soc. 362, 1445–1473 (2010)
Bodnarchuk, V., Kaluzhnin, L., Kotov, V., Romov, B.: Galois theory for Post algebras. I. Cybern. 5(3), 243–252 (1969)
Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra, graduate texts in mathematics edn. Springer, New York (2012). The Millenium Edition (electronic book) http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html.
Freese, R., Kiss, E., Valeriote, M.: Universal algebra calculator (2011). Available at: www.uacalc.org
Freese, R., Valeriote, M.: On the complexity of some Maltsev conditions. Int. J. Algebra Comput. 19(01), 41–77 (2009)
Geiger, D.: Closed systems of functions and predicates. Pacific J. Math. 27(1), 95–100 (1968)
Horowitz, J.: Computational complexity of various Mal’cev conditions. Int. J. Algebra Comput. 23(06), 1521–1531 (2013)
Idziak, P., Marković, P., McKenzie, R., Valeriote, M., Willard, R.: Tractability and learnability arising from algebras with few subpowers. SIAM J. Comput. 39(7), 3023–3037 (2010)
Kearnes, K.A., Szendrei, Á.: Clones of algebras with parallelogram terms. Int. J. Algebra Comput. 22(01), 1250,005 (2012)
Kearnes, K.A., Szendrei, Á.: Cube term blockers without finiteness. Algebra Univ. 78(4), 437–459 (2017)
Lindner, C.C., Rodger, C.A.: Design theory, 2nd edn. Chapman & Hall/CRC Press, Roca Baton, London, New York (2008)
Marković, P., Maróti, M., McKenzie, R.: Finitely related clones and algebras with cube terms. Order 29(2), 345–359 (2012)
Maróti, M.: The existence of a near-unanimity term in a finite algebra is decidable. J. Symb. Log. 74, 1001–1014 (2009)
Sequeira, L.: Near-unanimity is decomposable. Algebra Univ. 50(2), 157–164 (2003)
Zhuk, D.N.: Key (critical) relations preserved by a weak near-unanimity function. Algebra Univ. 77(2), 191–235 (2017)
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Our thanks go to Ralph Freese who has, with amazing speed, implemented Algorithm 1 in UAcalc.
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The first author was supported by the PRIMUS/SCI/12 and UNCE/SCI/22 projects of the Charles University and by the the Czech Science Foundation project GA ČR 18-20123S. The second author was supported by Russian Foundation for Basic Research (Grant 19-01-00200)
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Kazda, A., Zhuk, D. Existence of cube terms in finite algebras. Algebra Univers. 82, 11 (2021). https://doi.org/10.1007/s00012-020-00700-7
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DOI: https://doi.org/10.1007/s00012-020-00700-7