| Galois Connections and Applications 2004 Edition Contributor(s): Denecke, K. (Editor), Erné, M. (Editor), Wismath, S. L. (Editor) |
|
 |
ISBN: 1402018975 ISBN-13: 9781402018978 Publisher: Springer OUR PRICE: $104.49 Product Type: Hardcover - Other Formats Published: March 2004 Annotation: This book presents the main ideas of General Galois Theory as a generalization of Classical Galois Theory. It sketches the development of Galois connections through the last three centuries. 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. |
| Additional Information |
| BISAC Categories: - Mathematics | Algebra - Abstract - Mathematics | Group Theory - Mathematics | Algebra - General |
| Dewey: 512.32 |
| LCCN: 2004041011 |
| Series: Mathematics and Its Applications |
| Physical Information: 1.04" H x 6.26" W x 9.66" (2.31 lbs) 502 pages |
| Descriptions, Reviews, Etc. |
| Publisher 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". |