Galois Connections with Truth Stressers: Foundations for Formal Concept Analysis of Object-Attribute Data with Fuzzy Attributes

  • Radim Bělohlávek
  • Taťána Funioková
  • Vilém Vychodil
Part of the Advances in Soft Computing book series (AINSC, volume 33)


Galois connections appear in several areas of mathematics and computer science, and their applications. A Galois connection between sets X and Y is a pair 〈 ↑, ↓〉 of mappings ↑ assigning subcollections of Y to subcollections of X, and ↓ assigning subcollections of X to subcollections of Y. By definition, Galois connections have to satisfy certain conditions. Galois connections can be interpreted in the following manner: For subcollections A and B of X and Y, respectively, A↑ is the collection of all elements of Y which are in a certain relationship to all elements from A, and B↓ is the collection of all elements of X which are in the relationship to all elements in B. From the very many examples of Galois connections in mathematics, let us recall the following. Let X be the set of all logical formulas of a given language, Y be the set of all structures (interpretations) of the same language. For AX and BY, let A↑ consist of all structures in which each formula from A is true, let B↓ denote the set of all formulas which are true in each structure from B. Then, ↑ and ↓ is a Galois connection.


Fuzzy Logic Truth Stresser Formal Concept Residuated Lattice Concept Lattice 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Radim Bělohlávek
    • 1
    • 2
  • Taťána Funioková
    • 1
  • Vilém Vychodil
    • 1
  1. 1.Dept. Computer SciencePalacky UniversityOlomoucCzech Republic
  2. 2.Inst. Research and Applications of Fuzzy ModelingUniversity of OstravaOstravaCzech Republic

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