Knowledge description and Galois correspondence

Auteurs

• Hermann Rodenhausen

Paramètres

• Mathématiques

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The concept of a Galois connection is a vital tool for analyzing relationships between various order structures. It was implicitly utilized by Galois in his theory of algebraic equations, where the relation between subgroups of the Galois group and intermediate fields can be described by a Galois connection. In logic, this concept relates structures and sentences through validity relations. In computer science and formal linguistics, it manifests in the notion of "context" in formal concept analysis, leading to the construction of Galois connections from binary relations interpreted as "incidence" between objects and attributes. The book explores how Galois connections differ from binary relations by proving a more general representation theorem. Each Galois connection includes natural closure operators, which often have appealing algebraic descriptions. The principles determining closure outcomes are analyzed using logic, establishing a connection between closure concepts across different algebraic categories. The relationship between algebraic closure forms and logical entailment is examined. Additionally, the concept of a "basis" in partial orders is addressed, demonstrating the existence and uniqueness of bases, which can be characterized by Galois connections. The book also extends the application of knowledge space theory, linking it to the study of mathematical misconceptions, revealing that conceptual failures exhibit

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• Année de publication: 2011