On the General Coloring Problem

  • N. W. Sauer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6570)


Generalizing relational structures and formal languages to structures whose relations are evaluated by elements of a lattice, we show that such structure classes form a Heyting algebra if and only if the evaluation lattice is a Heyting algebra. Hence various new and some older results obtained for Heyting algebras can be applied to such structure classes.


Generalized homomorphisms Heyting algebras 


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  1. 1.
    Maurer, H.A., Sudborough, J.H., Welzl, E.: On the complexity of the general coloring problem. Inform. and Control 51, 123–145 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Maurer, H.A., Salomaa, A., Wood, D.: Colorings and interpretations: a connection between graphs and grammar forms. Discrete Appl. Math. 3, 119–135 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Maurer, H.A., Salomaa, A., Wood, D.: Dense hierarchies of grammatical families. J. ACM 29(1), 118–126 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Maurer, H.A., Salomaa, A., Wood, D.: Dense hierarchies of grammatical families. J. Assoc. Comput. Mach. 29(1), 118–126 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duffus, D., Sauer, N.: Lattices arising in categorical investigations of Hedetniemi’s conjecture. Discrete Math. 152, 125–139 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974)zbMATHGoogle Scholar
  7. 7.
    Birkhoff, G.: Generalized arithmetic. Duke Math. J. 12, 283–302 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Rutherford, D.E.: Introduction to Lattice Theory. Oliver and Boyd (1965)Google Scholar
  9. 9.
    Gierz, G., Hoffmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications 93 (2003)Google Scholar
  10. 10.
    Kuich, W., Sauer, N., Urbanek, F.: Heyting Algebras and Formal Languages. J. of Universal Computer Science 8(7), 722–736 (2002)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Tardif, C.: Hedetniemi’s conjecture, 40 years later. Graph Theory Notes of New York LIV, pp. 46–57. New York Academy of Sciences (2008)Google Scholar
  12. 12.
    Zhu, X.: A survey on Hedetniemi’s conjecture. Taiwanese Journal of Mathematics 2(1), 1–24 (1998)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Sauer, N.: Hedetniemis Conjecture–a survey. Combinatorics, graph theory, algorithms and applications. Discrete Math. 229(1-3), 261–292 (2001)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Foniok, J., Nešetřil, J., Pultr, A., Tardif, C.: Dualities and Dual Pairs in Heyting Algebras. Order. arXiv:0908.0428v1 (July 16, 2010)Google Scholar
  15. 15.
    Foniok, J., Nešetřil, J., Tardif, C.: Generalised dualities and maximal finite antichains in the homomorphism order of relational structures. European J. Combin. 29(4), 881–899 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nešetřil, J., Pultr, A., Tardif, C.: Gaps and dualities in Heyting categories. Comment. Math. Univ. Carolin. 48(1), 9–23 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Nešetřil, J., Tardif, C.: Duality theorems for finite structures (characterising gaps and good characterisations). J. Combin. Theory Ser. B 80(1), 80–97 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • N. W. Sauer
    • 1
  1. 1.University of Calgary and Technische Universität WienCanada

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