Abstract
The blocks of a tolerance relation generalize equivalence classes of equivalence relations and are in one to one correspondence with certain set coverings. We will, within the framework of formal concept analysis, analyse the blocks of the direct product of two tolerance relations. The question is how blocks of the direct product are related to the structure of the factors. It turns out that directly induced and non-induced blocks exist.
For tolerance relations, the problem of detecting the blocks of the direct product can be seen as a special instance of the task to determine the blocks of the union of two tolerance relations. In general, the blocks of the union of two tolerance relations are not directly derived from blocks of the unions components. Furthermore, we will apply our results to factor analysis and discuss open problems.
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Generally: \(X\subseteq Y^I\Leftrightarrow Y\subseteq X^I\Leftrightarrow X\times Y\subseteq I\).
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Acknowledgments
Finally, we want to express our thanks to the anonymous referees for their valuable suggestions to improve our paper.
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Jäkel, C., Schmidt, S.E. (2017). Blocks of the Direct Product of Tolerance Relations. In: Kryszkiewicz, M., Appice, A., Ślęzak, D., Rybinski, H., Skowron, A., Raś, Z. (eds) Foundations of Intelligent Systems. ISMIS 2017. Lecture Notes in Computer Science(), vol 10352. Springer, Cham. https://doi.org/10.1007/978-3-319-60438-1_58
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DOI: https://doi.org/10.1007/978-3-319-60438-1_58
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