The Tensor Product as a Lattice of Regular Galois Connections

  • Markus Krötzsch
  • Grit Malik
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3874)


Galois connections between concept lattices can be represented as binary relations on the context level, known as dual bonds. The latter also appear as the elements of the tensor product of concept lattices, but it is known that not all dual bonds between two lattices can be represented in this way. In this work, we define regular Galois connections as those that are represented by a dual bond in a tensor product, and characterize them in terms of lattice theory. Regular Galois connections turn out to be much more common than irregular ones, and we identify many cases in which no irregular ones can be found at all. To this end, we demonstrate that irregularity of Galois connections on sublattices can be lifted to superlattices, and observe close relationships to various notions of distributivity. This is achieved by combining methods from algebraic order theory and FCA with recent results on dual bonds. Disjunctions in formal contexts play a prominent role in the proofs and add a logical flavor to our considerations. Hence it is not surprising that our studies allow us to derive corollaries on the contextual representation of deductive systems.


Tensor Product Direct Product Propositional Logic Complete Lattice Deductive System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  2. 2.
    Krötzsch, M., Hitzler, P., Zhang, G.Q.: Morphisms in context. In: Conceptual Structures: Common Semantics for Sharing Knowledge. In: Proceedings of the 13th International Conference on Conceptual Structures, ICCS 2005, Kassel, Germany (2005), Extended version available at
  3. 3.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  4. 4.
    Ganter, B.: Relational Galois connections (2004) (unpublished manuscript)Google Scholar
  5. 5.
    Wille, R.: Tensor products of complete lattices as closure systems. Contributions to General Algebra 7, 381–386 (1991)MathSciNetGoogle Scholar
  6. 6.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. In: Encyclopedia of Mathematics and its Applications, vol. 93. Cambridge University Press, Cambridge (2003)Google Scholar
  7. 7.
    Krötzsch, M.: Morphisms in logic, topology, and formal concept analysis. Master’s thesis, Technische Universität Dresden (2005)Google Scholar
  8. 8.
    Barwise, J., Seligman, J.: Information flow: the logic of distributed systems. Cambridge tracts in theoretical computer science, vol. 44. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  9. 9.
    Johnstone, P.T.: Stone spaces. Cambridge University Press, Cambridge (1982)zbMATHGoogle Scholar
  10. 10.
    Goguen, J., Burstall, R.: Institutions: abstract model theory for specification and programming. Journal of the ACM 39 (1992)Google Scholar
  11. 11.
    Jung, A., Kegelmann, M., Moshier, M.A.: Multi lingual sequent calculus and coherent spaces. In: Fundamenta Informaticae, vol. XX, pp. 1–42 (1999)Google Scholar
  12. 12.
    Erné, M.: General Stone duality. Topology and its Applications 137, 125–158 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Markus Krötzsch
    • 1
  • Grit Malik
    • 2
  1. 1.AIFBUniversität KarlsruheGermany
  2. 2.Institut für AlgebraTechnische Universität DresdenGermany

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