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

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

Abstract

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.

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