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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:
03B52Fuzzy logic; logic of vagueness
06A15Galois correspondences (ordered structures)
06D72Fuzzy lattices etc.