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A note on variable threshold concept lattices: threshold-based operators are reducible to classical concept-forming operators. (English) Zbl 1119.06004

Summary: The present paper deals with formal concept analysis of data with fuzzy attributes. We clarify several points of a new approach of S. Q. Fan and W. X. Zhang [“Variable threshold concept lattice”, Inf. Sci. (to appear)], which is based on using thresholds in concept-forming operators. We show that the extent- and intent-forming operators of Fan and Zhang [loc. cit.] can be defined in terms of basic fuzzy set operations and the original operators as introduced and studied, e.g., by the present author [“Fuzzy Galois connections”, Math. Log. Q. 45, 497–504 (1999; Zbl 0938.03079); “Concept lattices and order in fuzzy logic”, Ann. Pure Appl. Logic 128, 277–298 (2004; Zbl 1060.03040)] and S. Pollandt [Fuzzy-Begriffe. Formale Begriffsanalyse unscharfer Daten. Berlin: Springer (1997; Zbl 0870.06008)]. As a consequence, the main properties of the new operators of Fan and Zhang [loc. cit.], including the properties studied by Fan and Zhang themselves, can be obtained as consequences of the original operators of the author [loc. cit.] and S. Pollandt [loc.cit.].

MSC:

06B99 Lattices
03B52 Fuzzy logic; logic of vagueness
06A15 Galois correspondences, closure operators (in relation to ordered sets)
68T30 Knowledge representation
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References:

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