## 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

Zbl 0491.06008
Full Text:

### 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). [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. · Zbl 1016.06008 [5] 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. [7] Ganter, B.; Wille, R., Formal concept analysis, mathematical foundations, (1999), 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 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 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. [16] Wille, R., Restructuring lattice theory: an approach based on hierarchies of concepts, (), 445-470 [17] Zadeh, L.A., Fuzzy sets, Inform. control, 8, 3, 338-353, (1965) · Zbl 0139.24606
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.