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International Conference on Formal Concept Analysis

ICFCA 2005: Formal Concept Analysis pp 269–284Cite as

Crisply Generated Fuzzy Concepts

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Part of the Lecture Notes in Computer Science book series (LNAI,volume 3403)

Abstract

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 ↓ (and so B = D ↓↑) for some crisp (i.e., ordinary) set D ⊆ 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).

Keywords

  • Fuzzy Logic
  • Formal Concept
  • Concept Lattice
  • Formal Concept Analysis
  • Fuzzy Concept

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Author information

Authors and Affiliations

  1. Dept. Computer Science, Palacký University, Tomkova 40, CZ-779 00, Olomouc, Czech Republic

    Radim Bělohlávek, Vladimír Sklenář & Jiří Zacpal

Authors
  1. Radim Bělohlávek
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  2. Vladimír Sklenář
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  3. Jiří Zacpal
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Editor information

Editors and Affiliations

  1. Institute of Algebra, University of Technology, Dresden, Germany

    Bernhard Ganter

  2. Département d’informatique, UQAM, P.O. Box, Montréal, (Qc), Canada

    Robert Godin

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Bělohlávek, R., Sklenář, V., Zacpal, J. (2005). Crisply Generated Fuzzy Concepts. In: Ganter, B., Godin, R. (eds) Formal Concept Analysis. ICFCA 2005. Lecture Notes in Computer Science(), vol 3403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32262-7_19

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  • DOI: https://doi.org/10.1007/978-3-540-32262-7_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24525-4

  • Online ISBN: 978-3-540-32262-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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