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Interval-valued fuzzy Galois connections: algebraic requirements and concept lattice construction. (English) Zbl 1205.68403

Summary: Fuzzy formal concept analysis is concerned with formal contexts expressing scalar-valued fuzzy relationships between objects and their properties. Existing fuzzy approaches assume that the relationship between a given object and a given property is a matter of degree in a scale \(L\) (generally [0,1]). However, the extent to which “object \(o\) has property \(a\)” may be sometimes hard to assess precisely. Then it is convenient to use a sub-interval from the scale \(L\) rather than a precise value. Such formal contexts naturally lead to interval-valued fuzzy formal concepts. The aim of the paper is twofold. We provide a sound minimal set of algebraic requirements for interval-valued implications in order to fulfill the fuzzy closure properties of the resulting Galois connection. Secondly, a new approach based on a generalization of Gödel implication is proposed for building the complete lattice of all interval-valued fuzzy formal concepts.

MSC:

68T30 Knowledge representation
03B52 Fuzzy logic; logic of vagueness
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B75 Generalizations of lattices
68T37 Reasoning under uncertainty in the context of artificial intelligence
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