Concept lattices and order in fuzzy logic (Q1877099)

The paper presents a generalization of the theory of concept lattices that were originated and further studied by R. Wille and his school [\textit{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 (\textbf{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.