基本説明
Examples of Galois connections as powerful tools in Category Theory and Universal Algebra are given. Applications of Galois connections in Linguistic and Data Analysis are presented.
Full Description
Galois connections provide the order- or structure-preserving passage between two worlds of our imagination - and thus are inherent in hu man thinking wherever logical or mathematical reasoning about cer tain hierarchical structures is involved. Order-theoretically, a Galois connection is given simply by two opposite order-inverting (or order preserving) maps whose composition yields two closure operations (or one closure and one kernel operation in the order-preserving case). Thus, the "hierarchies" in the two opposite worlds are reversed or transported when passing to the other world, and going forth and back becomes a stationary process when iterated. The advantage of such an "adjoint situation" is that information about objects and relationships in one of the two worlds may be used to gain new information about the other world, and vice versa. In classical Galois theory, for instance, properties of permutation groups are used to study field extensions. Or, in algebraic geometry, a good knowledge of polynomial rings gives insight into the structure of curves, surfaces and other algebraic vari eties, and conversely. Moreover, restriction to the "Galois-closed" or "Galois-open" objects (the fixed points of the composite maps) leads to a precise "duality between two maximal subworlds".
Contents
Adjunctions and Galois Connections: Origins, History and Development.- Categorical Galois Theory: Revision and Some Recent Developments.- The Polarity between Approximation and Distribution.- Galois Connections and Complete Sublattices.- Galois Connections for Operations and Relations.- Galois Connections and Polynomial Completeness.- Q-Independence and Weak Automorphisms.- A Survey of Clones Closed Under Conjugation.- Galois Connections for Partial Algebras.- Complexity of Terms and the Galois Connection Id-Mod.- Iterated Galois Connections in Arithmetic and Linguistics.- Deductive Systems and Galois Connections.- A Galois Correspondence for Digital Topology.- Galois Connections in Category Theory, Topology and Logic.- Dyadic Mathematics — Abstractions from Logical Thought.
