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Crisply generated fuzzy concepts. (English) Zbl 1078.68142
Ganter, Bernhard (ed.) et al., Formal concept analysis. Third international conference, ICFCA 2005, Lens, France, February 14–18, 2005. Proceedings. Berlin: Springer (ISBN 3-540-24525-1/pbk). Lecture Notes in Computer Science 3403. Lecture Notes in Artificial Intelligence, 269-284 (2005).
Summary: In formal concept analysis of data with fuzzy attributes, both the extent and the intent of a formal (fuzzy) concept may be fuzzy sets. In this paper we focus on so-called crisply generated formal concepts. A concept \(\langle A,B\rangle \in \mathcal B(X,Y,I)\) is crisply generated if \(A = D^\downarrow\) (and so \(B = D^{\downarrow\uparrow}\)) for some crisp (i.e., ordinary) set \(D \subseteq Y\) of attributes (generator). Considering only crisply generated concepts has two practical consequences. First, the number of crisply generated formal concepts is considerably less than the number of all formal fuzzy concepts. Second, since crisply generated concepts may be identified with a (ordinary, not fuzzy) set of attributes (the largest generator), they might be considered “the important ones” among all formal fuzzy concepts. We present basic properties of the set of all crisply generated concepts, an algorithm for listing all crisply generated concepts, a version of the main theorem of concept lattices for crisply generated concepts, and show that crisply generated concepts are just the fixed points of pairs of mappings resembling Galois connections. Furthermore, we show connections to other papers on formal concept analysis of data with fuzzy attributes. Also, we present examples demonstrating the reduction of the number of formal concepts and the speed-up of our algorithm (compared to listing of all formal concepts and testing whether a concept is crisply generated).
For the entire collection see [Zbl 1069.68007].

MSC:
68T30 Knowledge representation
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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