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Galois Connections and Applications

  • K. Denecke
  • M. Erné
  • S. L. Wismath

Part of the Mathematics and Its Applications book series (MAIA, volume 565)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Klaus Denecke, Shelly L. Wismath
    Pages 211-229
  3. Reinhard Pöschel
    Pages 231-258
  4. K. Glazek, S. Niwczyk
    Pages 277-295
  5. Ágnes Szendrei
    Pages 297-343
  6. Peter Burmeister
    Pages 345-370
  7. Klaus Denecke, Shelly L. Wismath
    Pages 371-388
  8. I. Chajda, R. Halaš
    Pages 399-411
  9. Back Matter
    Pages 499-501

About this book

Introduction

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".

Keywords

Algebra Arithmetic Category theory Morphism Topology calculus complexity linguistics mathematics

Editors and affiliations

  • K. Denecke
    • 1
  • M. Erné
    • 2
  • S. L. Wismath
    • 3
  1. 1.University of PotsdamPotsdamGermany
  2. 2.University of HannoverHannoverGermany
  3. 3.University of LethbridgeLethbridgeCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4020-1898-5
  • Copyright Information Springer Science+Business Media Dordrecht 2004
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-6540-7
  • Online ISBN 978-1-4020-1898-5
  • Buy this book on publisher's site