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A categorical view at generalized concept lattices. (English) Zbl 1132.06300
Summary: We continue in the direction of the ideas from G.-Q. Zhang’s paper [“Chu spaces, concept lattices, and domains”, in: Proc. 19th conference on mathematical foundations of programming semantics, Montreal 2003, Electron. Notes Theor. Comput. Sci. 83 (2004)] about a relationship between Chu spaces and formal concept analysis. We modify this categorical point of view at a classical concept lattice to a generalized concept lattice (in the sense of [S. Krajči, “A generalized concept lattice”, Log. J. IGPL 13, No. 5, 543–550 (2005; Zbl 1088.06005)]): We define generalized Chu spaces and show that they together with (a special type of) their morphisms form a category. Moreover we define corresponding modifications of the image/inverse image operator and show their commutativity properties with mapping defining generalized concept lattice as fuzzifications of Zhang’s ones.

MSC:
06B23 Complete lattices, completions
06D72 Fuzzy lattices (soft algebras) and related topics
68T30 Knowledge representation
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[1] Bělohlávek R.: Fuzzy concepts and conceptual structures: induced similarities. Proc. Joint Conference Inform. Sci. ’98, Durham (U.S.A.) 1998, Vol. I, pp. 179-182
[2] Bělohlávek R.: Concept lattices and order in fuzzy logic. Ann. Pure Appl. Logic 128 (2004), 277-298 · Zbl 1060.03040 · doi:10.1016/j.apal.2003.01.001
[3] Bělohlávek R., Sklenář, V., Zacpal J.: Crisply generated fuzzy concepts. ICFCA 2005 (B. Ganter and R. Godin, Lecture Notes in Computer Science 3403), Springer-Verlag, Berlin - Heidelberg 2005, pp. 268-283 · Zbl 1078.68142 · doi:10.1007/b105806
[4] Bělohlávek R., Vychodil V.: Reducing the size of fuzzy concept lattices by hedges. Proc. FUZZ-IEEE 2005, The IEEE Internat. Conference Fuzzy Systems, Reno 2005, pp. 663-668
[5] Yahia S. Ben, Jaoua A.: Discovering knowledge from fuzzy concept lattice. Data Mining and Computational Intelligence (A. Kandel, M. Last, and H. Bunke, Physica-Verlag, Heidelberg 2001, pp. 169-190
[6] Ganter B., Wille R.: Formal Concept Analysis, Mathematical Foundation. Springer-Verlag, Berlin 1999 · Zbl 0909.06001
[7] Krajči S.: A generalized concept lattice. Logic J. of the IGPL 13 (2005), 5, 543-550 · Zbl 1088.06005 · doi:10.1093/jigpal/jzi045
[8] Krajči S.: The basic theorem on generalized concept lattice. Proc. 2nd Internat. Workshop CLA 2004 (V. Snášel and R. Bělohlávek, Ostrava 2004, pp. 25-33
[9] Krajči S.: Every concept lattice with hedges is isomorphic to some generalized concept lattice. Proc. 3nd Internat. Workshop CLA 2004 (R. Bělohlávek and V. Snášel, Olomouc 2005, pp. 1-9
[10] Krajči S.: Cluster based efficient generation of fuzzy concepts. Neural Network World 13 (2003), 5, 521-530
[11] Pollandt S.: Fuzzy Begriffe. Springer-Verlag, Berlin 1997 · Zbl 0870.06008
[12] Pollandt S.: Datenanalyse mit Fuzzy-Begriffen. Begriffliche Wissensverarbeitung, Methoden und Anwendungen (G. Stumme and R. Wille, Springer-Verlag, Heidelberg 2000, pp. 72-98 · Zbl 0958.68162
[13] Shostak A.: Fuzzy categories versus categories of fuzzily structured sets: Elements of the theory of fuzzy categories. Mathematik-Arbeitspapiere N 48: Categorical Methods in Algebra and Topology (A collection of papers in honor of Horst Herrlich, Hans-E. Porst, Bremen 1977, pp. 407-438
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