The Tensor Product as a Lattice of Regular Galois Connections

Markus KrötzschMarkus Krötzsch, Grit MalikGrit Malik

ISBN: 3-540-32203-5
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Markus Krötzsch, Grit Malik
The Tensor Product as a Lattice of Regular Galois Connections
In Rokia Missaoui and Jürg Schmid, eds., Proceedings of the 4th International Conference on Formal Concept Analysis (ICFCA2006), Dresden, Germany, volume 3874 of Lecture Notes in Computer Science, February 2006. Springer

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KurzfassungAbstract
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 <i>regular</i> 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.
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Forschungsgruppe:Research Group: Wissensbasierte SystemeKnowledge-Based Systems

The final publication is available at Springer.

@inproceedings{KM2006,
  author    = {Markus Kr{\"{o}}tzsch and Grit Malik},
  title     = {The Tensor Product as a Lattice of Regular Galois Connections},
  editor    = {Rokia Missaoui and J{\"{u}}rg Schmid},
  booktitle = {Proceedings of the 4th International Conference on Formal Concept
               Analysis (ICFCA2006), Dresden, Germany},
  series    = {Lecture Notes in Computer Science},
  volume    = {3874},
  publisher = {Springer},
  year      = {2006},
  month     = {February}
}