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Testing for a Semilattice Term

  • Ralph Freese
  • J. B. Nation
  • Matt Valeriote


This paper investigates the computational complexity of deciding if a given finite algebra is an expansion of a semilattice. In general this problem is known to be EXP-TIME complete, and we show that even for idempotent algebras, this problem remains hard. This result is in contrast to a series of results that show that similar decision problems turn out to be tractable.


Semilattice Computational complexity Maltsev condition Idempotent algebra 


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The first author was supported by the National Science Foundation under grant No. 1500235 and the third author was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.


  1. 1.
    Baker, K.A.: Congruence-distributive polynomial reducts of lattices. Algebra Univers. 9(1), 142–145 (1979)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barto, L., Krokhin, A., Willard, R.: Polymorphisms, and how to use them. In: Krokhin, A., Zivny, S. (eds.) The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pp. 1–44. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl (2017)Google Scholar
  3. 3.
    Bergman, C.: Universal Algebra, Volume 301 of Pure and Applied Mathematics (Boca Raton). CRC Press, Boca Raton (2012). Fundamentals and selected topicsGoogle Scholar
  4. 4.
    Bergman, C., Juedes, D., Slutzki, G.: Computational complexity of term-equivalence. Int. J. Algebra Comput. 9(1), 113–128 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra, Volume 78 of Graduate Texts in Mathematics. Springer, New York-Berlin (1981)MATHGoogle Scholar
  6. 6.
    Freese, R., Valeriote, M.A.: On the complexity of some Maltsev conditions. Int. J. Algebra Comput. 19(1), 41–77 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Freese, R., Kiss, E., Valeriote, M.: Universal Algebra Calculator. Available at: (2011)
  8. 8.
    Hobby, D., McKenzie, R.: The Structure of Finite Algebras, Volume 76 of Contemporary Mathematics. American Mathematical Society, Providence (1988). Revised edition: 1996CrossRefMATHGoogle Scholar
  9. 9.
    Horowitz, J.: Computational complexity of various Mal’cev conditions. Int. J. Algebra Comput. 23(6), 1521–1531 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jónsson, B.: Algebras whose congruence lattices are distributive. Math. Scand. 21(1968), 110–121 (1967)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kazda, A., Valeriote, M.: Deciding some Maltsev conditions in finnite idempotent algebras. Preprint (2017)Google Scholar
  12. 12.
    Kearnes, K.A., Kiss, E.W.: The shape of congruence lattices. Mem. Am. Math. Soc. 222(1046), viii+ 169 (2013)MathSciNetMATHGoogle Scholar
  13. 13.
    Taylor, W.: Simple equations on real intervals. Algebra Univers. 61(2), 213–226 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Valeriote, M., Willard, R.: Idempotent n-permutable varieties. Bull. Lond. Math. Soc. 46(4), 870–880 (2014)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Hawaii at ManoaHonoluluUSA
  2. 2.McMaster UniversityHamiltonCanada

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