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

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 A ⊆ X and B ⊆ Y, 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.