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Concept lattices and order in fuzzy logic. (English) Zbl 1060.03040

The paper presents a generalization of the theory of concept lattices that were originated and further studied by R. Wille and his school [R. Wille, “Restructuring lattice theory: an approach based on hierarchies of concepts”, in: Ordered sets, Proc. NATO Adv. Study Inst., Banff/Can. 1981, 445–470 (1982; Zbl 0491.06008)]. The theory is based on a generalization to the structure of truth values forming a residuated lattice, where the adjointness condition is an algebraic counterpart of the many-valued modus ponens rule of fuzzy logic.
In the paper, the notions of fuzzy partial order (L-order) with respect to some fuzzy equality relation, lattice order, and fuzzy formal concepts are studied. The main result is a theorem characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context. The paper ends with a theorem on Dedekind-MacNeille completion for fuzzy orders.

MSC:

03B52 Fuzzy logic; logic of vagueness
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06D72 Fuzzy lattices (soft algebras) and related topics

Citations:

Zbl 0491.06008
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References:

[1] A. Arnauld, P. Nicole, La logique ou l’art de penser, 1662 (Also in German: Die Logik oder die Kunst des Denkens, Darmstadt, 1972).; A. Arnauld, P. Nicole, La logique ou l’art de penser, 1662 (Also in German: Die Logik oder die Kunst des Denkens, Darmstadt, 1972).
[2] Banaschewski, B., Hüllensysteme und Erweiterungen von Quasiordnungen, Z. Math. Logic Grundlagen Math., 2, 117-130 (1956) · Zbl 0073.26904
[3] Bělohlávek, R., Fuzzy Galois connections, Math. Logic Quart., 45, 4, 497-504 (1999) · Zbl 0938.03079
[4] R. Bělohlávek, Reduction and a simple proof of characterization of fuzzy concept lattices, Fund. Inform. 46 (4) (2001) 277-285.; R. Bělohlávek, Reduction and a simple proof of characterization of fuzzy concept lattices, Fund. Inform. 46 (4) (2001) 277-285. · Zbl 1016.06008
[5] U. Bodenhofer, A similarity-based generalization of fuzzy orderings, Ph.D. Thesis, Universitätsverlag R. Trauner, Linz, 1999.; U. Bodenhofer, A similarity-based generalization of fuzzy orderings, Ph.D. Thesis, Universitätsverlag R. Trauner, Linz, 1999. · Zbl 0949.03049
[6] G. Birkhoff, Lattice Theory, 3rd Edition, AMS Coll. Publ., vol. 25, American Mathematical Society, Providence, RI, 1967.; G. Birkhoff, Lattice Theory, 3rd Edition, AMS Coll. Publ., vol. 25, American Mathematical Society, Providence, RI, 1967.
[7] Ganter, B.; Wille, R., Formal Concept Analysis, Mathematical Foundations (1999), Springer: Springer Berlin · Zbl 0909.06001
[8] Goguen, J. A., L-fuzzy sets, J. Math. Anal. Appl., 18, 145-174 (1967) · Zbl 0145.24404
[9] Hájek, P., Metamathematics of Fuzzy Logic (1998), Kluwer: Kluwer Dordrecht · Zbl 0937.03030
[10] Höhle, U., On the fundamentals of fuzzy set theory, J. Math. Anal. Appl., 201, 786-826 (1996) · Zbl 0860.03038
[11] MacNeille, H. M., Partially ordered sets, Trans. Amer. Math. Soc., 42, 416-460 (1937) · Zbl 0017.33904
[12] Ore, O., Galois connections, Trans. Amer. Math. Soc., 55, 493-513 (1944)
[13] Pollandt, S., Fuzzy Begriffe (1997), Springer: Springer Berlin · Zbl 0870.06008
[14] Schmidt, J., Zur Kennzeichnung der Dedekind-MacNeillschen Hülle einer geordneten Menge, Arch. Math., 7, 241-249 (1956) · Zbl 0073.03801
[15] E. Schröder, Algebra der Logik I, II, III, Leipzig, 1890, 1891, 1895.; E. Schröder, Algebra der Logik I, II, III, Leipzig, 1890, 1891, 1895.
[16] Wille, R., Restructuring lattice theory: an approach based on hierarchies of concepts, (Rival, I., Ordered Sets (1982), Reidel, Dordrecht: Reidel, Dordrecht Boston), 445-470
[17] Zadeh, L. A., Fuzzy sets, Inform. Control, 8, 3, 338-353 (1965) · Zbl 0139.24606
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