Galois Connections and Applications:
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Dordrecht
Springer Netherlands
2004
|
| Series: | Mathematics and Its Applications
565 |
| Subjects: | |
| Online Access: | Volltext |
| Item 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" |
| Physical Description: | 1 Online-Ressource (XVI, 502 p) |
| ISBN: | 9781402018985 9789048165407 |
| DOI: | 10.1007/978-1-4020-1898-5 |
Staff View
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| 500 | |a 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" | ||
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| id | DE-604.BV042419231 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:04Z |
| institution | BVB |
| isbn | 9781402018985 9789048165407 |
| language | English |
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| spelling | Denecke, K. Verfasser aut Galois Connections and Applications edited by K. Denecke, M. Erné, S. L. Wismath Dordrecht Springer Netherlands 2004 1 Online-Ressource (XVI, 502 p) txt rdacontent c rdamedia cr rdacarrier Mathematics and Its Applications 565 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" Mathematics Computer science Data structures (Computer science) Artificial intelligence Mathematics, general Programming Languages, Compilers, Interpreters Artificial Intelligence (incl. Robotics) Data Structures Informatik Künstliche Intelligenz Mathematik Erné, M. Sonstige oth Wismath, S. L. Sonstige oth https://doi.org/10.1007/978-1-4020-1898-5 Verlag Volltext |
| spellingShingle | Denecke, K. Galois Connections and Applications Mathematics Computer science Data structures (Computer science) Artificial intelligence Mathematics, general Programming Languages, Compilers, Interpreters Artificial Intelligence (incl. Robotics) Data Structures Informatik Künstliche Intelligenz Mathematik |
| title | Galois Connections and Applications |
| title_auth | Galois Connections and Applications |
| title_exact_search | Galois Connections and Applications |
| title_full | Galois Connections and Applications edited by K. Denecke, M. Erné, S. L. Wismath |
| title_fullStr | Galois Connections and Applications edited by K. Denecke, M. Erné, S. L. Wismath |
| title_full_unstemmed | Galois Connections and Applications edited by K. Denecke, M. Erné, S. L. Wismath |
| title_short | Galois Connections and Applications |
| title_sort | galois connections and applications |
| topic | Mathematics Computer science Data structures (Computer science) Artificial intelligence Mathematics, general Programming Languages, Compilers, Interpreters Artificial Intelligence (incl. Robotics) Data Structures Informatik Künstliche Intelligenz Mathematik |
| topic_facet | Mathematics Computer science Data structures (Computer science) Artificial intelligence Mathematics, general Programming Languages, Compilers, Interpreters Artificial Intelligence (incl. Robotics) Data Structures Informatik Künstliche Intelligenz Mathematik |
| url | https://doi.org/10.1007/978-1-4020-1898-5 |
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