Isotone fuzzy Galois connections with hedges. (English) Zbl 1226.06001

Summary: We study isotone fuzzy Galois connections and concept lattices parameterized by particular unary operators. The operators represent linguistic hedges such as “very”, “rather”, “more or less”, etc. Isotone fuzzy Galois connections and concept lattices provide an alternative to their antitone counterparts which are the fundamental structures behind formal concept analysis of data with fuzzy attributes. We show that hedges enable us to control the number of formal concepts in the associated concept lattice. We also describe the structure of the concept lattice and provide a counterpoint to the main theorem of concept lattices.


06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B99 Lattices
68T30 Knowledge representation
Full Text: DOI


[1] E. Bartl, R. Belohlavek, J. Konecny, V. Vychodil, Isotone Galois connections and concept lattices with hedges. in: IEEE IS 2008, Varna, Bulgaria, pp. 15-24-15-28.
[2] Belohlavek, R., Fuzzy Galois connections, Math. logic quart., 45, 4, 497-504, (1999) · Zbl 0938.03079
[3] Belohlavek, R., Fuzzy closure operators, J. math. anal. appl., 262, 473-489, (2001) · Zbl 0989.54006
[4] Belohlavek, R., Reduction and a simple proof of characterization of fuzzy concept lattices, Fundam. inform., 46, 4, 277-285, (2001) · Zbl 1016.06008
[5] Belohlavek, R., Fuzzy relational systems: foundations and principles, (2002), Kluwer, Academic/Plenum Publishers New York · Zbl 1067.03059
[6] Belohlavek, R., Concept lattices and order in fuzzy logic, Ann. pure appl. logic, 128, 277-298, (2004) · Zbl 1060.03040
[7] Belohlavek, R.; Funiokova, T.; Vychodil, V., Fuzzy closure operators with truth stressers, Logic J. IGPL, 13, 5, 503-513, (2005) · Zbl 1089.06001
[8] R. Belohlavek, V. Vychodil, Reducing the size of fuzzy concept lattices by hedges, in: FUZZ-IEEE 2005, The IEEE International Conference on Fuzzy Systems, Reno (Nevada, USA), May 22-25, 2005, pp. 663-668.
[9] Belohlavek, R.; Vychodil, V., Fuzzy concept lattices constrained by hedges, Jaciii, 11, 6, 536-545, (2007)
[10] R. Belohlavek, Optimal decompositions of matrices with grades, in: IEEE IS 2008, Proceedings of the International IEEE Conference on Intelligent Systems 2008, Varna, Bulgaria, 2008, pp. 15-2-15-7.
[11] R. Belohlavek, J. Konecny, A logic of attribute containment, in: KAM’08, Wuhan, China, 2008, pp. 246-251.
[12] Belohlavek, R., Optimal triangular decompositions of matrices with entries from residuated lattices, Int. J. approx. reason., 50, 8, 1250-1258, (2009) · Zbl 1195.15011
[13] I. Düntsch, G. Gediga, Modal-style operators in qualitative data analysis, in: IEEE International Conference on Data Mining, 2002, pp. 155-163, ISBN:0-7695-1754-4.
[14] Ganter, B.; Wille, R., Formal concept analysis, Mathematical foundations, (1999), Springer-Verlag Berlin
[15] Georgescu, G.; Popescu, A., Non-dual fuzzy connections, Arch. math. logic, 43, (2004)
[16] Hájek, P., Methamathematics of fuzzy logic, (1998), Kluwer Dordrecht
[17] Hájek, P., On very true, Fuzzy sets syst., 124, 329-333, (2001) · Zbl 0997.03028
[18] Ore, O., Galois connexions, Trans. am. math. soc., 55, 493-513, (1944) · Zbl 0060.06204
[19] Polandt, S., Fuzzy begriffe, (1977), Springer-Verlag Berlin/Heidelberg
[20] Popescu, A., A general approach to fuzzy concepts, Math. logic quart., 50, 1-17, (2004)
[21] Vychodil, V., Truth-depressing hedges and BL-logic, Fuzzy sets syst., 157, 15, 2074-2090, (2006) · Zbl 1114.03023
[22] Ward, M.; Dilworth, R.P., Residuated lattices, Trans. am. math. soc., 45, 335-354, (1939) · Zbl 0021.10801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.