Abstract
It is a well-known fact that complete tolerance relations on concept lattices are in one-to-one correspondence with some superrelations (called block relations) of the incidence relation of the underlying formal context. However, sometimes it is useful to consider more general superrelations of the incidence relation, leading to tolerance relations, not compatible with the lattice structure of the concept lattice. In this paper, we give a characterization of such tolerances and present a mathematical framework for factorizing any complete lattice by such incompatible tolerances.
Supported by grant no. P103/10/1056 of the Czech Science Foundation.
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References
Belohlavek, R.: Concept lattices and order in fuzzy logic. Ann. Pure Appl. Log. 128(1-3), 277–298 (2004)
Belohlavek, R.: Similarity relations in concept lattices. J. Log. Comput. 10(6), 823–845 (2000)
Belohlavek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publishers, Norwell (2002)
Cheung, K., Vogel, D.: Complexity reduction in lattice-based information retrieval. Inf. Retr. 8(2), 285–299 (2005)
Czédli, G.: Factor lattices by tolerances. Acta Sci. Math. 44, 35–42 (1982)
Ganter, B., Wille, R.: Formal Concept Analysis – Mathematical Foundations. Springer, Heidelberg (1999)
Harding, J.: Completions of ordered algebraic structures: A survey. In: Huynh, V.N., et al. (eds.) Interval / Probabilistic Uncertainty and Non-Classical Logics. AISC, vol. 46, pp. 231–244. Springer, Heidelberg (2008)
Krupka, M.: An alternative version of the main theorem of fuzzy concept lattices. In: Trappl, R. (ed.) Cybernetics and Systems 2010, pp. 9–14. Austrian Society for Cybernetic Studies, Vienna (2010) (extended version submitted)
Krupka, M.: Factorization of residuated lattices. Log. J. IGPL 17(2), 205–223 (2009)
Snelting, G.: Concept lattices in software analysis. In: Ganter, B., Stumme, G., Wille, R. (eds.) Formal Concept Analysis. LNCS (LNAI), vol. 3626, pp. 272–287. Springer, Heidelberg (2005)
Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, Boston, pp. 445–470 (1982); seminal publication on Formal Concept Analysis
Wille, R.: Complete tolerance relations of concept lattices. In: Eigenthaler, G., et al. (eds.) Contributions to General Algebra, vol. 3, pp. 397–415. Hölder-Pichler-Tempsky, Wien (1985)
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Krupka, M. (2011). On Factorization of Concept Lattices by Incompatible Tolerances. In: Valtchev, P., Jäschke, R. (eds) Formal Concept Analysis. ICFCA 2011. Lecture Notes in Computer Science(), vol 6628. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20514-9_14
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DOI: https://doi.org/10.1007/978-3-642-20514-9_14
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