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Finite bands are finitely related

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Abstract

We prove that every finite idempotent semigroup (band) is finitely related, which means that the clone of its term operations (i.e. operations induced by words) is determined by finitely many relations. This solves an open problem posed by Mayr (Semigroup Forum 86:613–633, 2013).

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References

  1. Aichinger, E.: Constantive Mal’cev clones on finite sets are finitely related. Proc. Am. Math. Soc. 138, 3501–3507 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aichinger, E., Mayr, P., McKenzie, R.: On the number of finite algebraic structures. J. Eur. Math. Soc. 16, 1673–1686 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barto, L.: Finitely related algebras in congruence distributive varieties have near unanimity terms. Can. J. Math. 65, 3–21 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barto, L.: Finitely related algebras in congruence modular varieties have few subpowers. J. Eur. Math. Soc. (to appear). http://www.karlin.mff.cuni.cz/~barto/Articles/ValerioteConjecture.pdf

  5. Berman, J., Idziak, P., Marković, P., McKenzie, R., Valeriote, M., Willard, R.: Varieties with few subalgebras of powers. Trans. Am. Math. Soc. 362, 1445–1473 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Biryukov, A.P.: Varieties of idempotent semigroups. Algebra i Logika 9, 255–273 (1970). (in Russian)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 720–742 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra, Graduate Texts in Mathematics. Springer-Verlag, New York (1981)

    MATH  Google Scholar 

  9. Davey, B.A., Jackson, M.G., Pitkethly, J.G., Szabó, Cs: Finite degree: algebras in general and semigroups in particular. Semigroup Forum 83, 89–110 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fennemore, C.F.: All varieties of bands. I, II. Math. Nachr. 48, 237–252, 253–262 (1971)

  11. Gerhard, J.A.: The lattice of equational classes of idempotent semigroups. J. Algebra 15, 195–224 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gerhard, J.A., Petrich, M.: Free bands and free \(^\ast \)-bands. Glasgow Math. J. 28, 161–179 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gerhard, J .A., Petrich, M.: Varieties of bands revisited. Proc. Lond. Math. Soc. 3(58), 323–350 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jablonskiĭ, S.V.: The structure of the upper neighbourhood for predicately describable classes in \(P_k\). Dokl. Akad. Nauk SSSR 218, 304–307 (1974) (in Russian). English traslation: Sov. Math. Dokl. 15, 1353–1356 (1974)

  15. Jackson, M. (ed.).: General Algebra and Its Applications 2013: Problem Session. Algebra Univ. 74, 9–16 (2015)

  16. Marković, P., Maróti, M., McKenzie, R.: Finitely related clones and algebras with cube terms. Order 29, 345–359 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mayr, P.: On finitely related semigroups. Semigroup Forum 86, 613–633 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Romov, B.A.: Local characterizations of Post algebras, I. Kibernetika 5, 38–45 (1976)

    MathSciNet  MATH  Google Scholar 

  19. Rosenberg, I.G., Szendrei, Á.: Degrees of clones and relations. Houston J. Math. 9, 545–580 (1983)

    MathSciNet  MATH  Google Scholar 

  20. Sapir, O.: The variety of idempotent semigroups is inherently non-finitely generated. Semigroup Forum 71, 140–146 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is grateful to a thorough and perceptive referee for a handful of comments which significantly improved the paper.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad, 21101, Serbia

    Igor Dolinka

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  1. Igor Dolinka
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Correspondence to Igor Dolinka.

Additional information

Communicated by Mikhail Volkov.

This research was supported by the Ministry of Education, Science, and Technological Development of the Republic of Serbia through the Grant No. 174019.

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Dolinka, I. Finite bands are finitely related. Semigroup Forum 97, 115–130 (2018). https://doi.org/10.1007/s00233-017-9897-y

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  • DOI: https://doi.org/10.1007/s00233-017-9897-y

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