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Galois Connections and Applications

  • Book
  • © 2004

Overview

Editors:
  1. K. Denecke

    1. University of Potsdam, Potsdam, Germany

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    You can also search for this editor in PubMed   Google Scholar

  2. M. Erné

    1. University of Hannover, Hannover, Germany

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    You can also search for this editor in PubMed   Google Scholar

  3. S. L. Wismath

    1. University of Lethbridge, Lethbridge, Canada

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  • The only book to describe the use of Galois connections in a wide field of branches of mathematics and outside of mathematics

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

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xv
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  2. Adjunctions and Galois Connections: Origins, History and Development

    • Marcel Erné
    Pages 1-138

  3. Categorical Galois Theory: Revision and Some Recent Developments

    • G. Janelidze
    Pages 139-171

  4. The Polarity between Approximation and Distribution

    • Marcel Erné
    Pages 173-210

  5. Galois Connections and Complete Sublattices

    • Klaus Denecke, Shelly L. Wismath
    Pages 211-229

  6. Galois Connections for Operations and Relations

    • Reinhard Pöschel
    Pages 231-258

  7. Galois Connections and Polynomial Completeness

    • K. Kaarli
    Pages 259-276

  8. Q-Independence and Weak Automorphisms

    • K. Glazek, S. Niwczyk
    Pages 277-295

  9. A Survey of Clones Closed Under Conjugation

    • Ágnes Szendrei
    Pages 297-343

  10. Galois Connections for Partial Algebras

    • Peter Burmeister
    Pages 345-370

  11. Complexity of Terms and the Galois Connection Id-Mod

    • Klaus Denecke, Shelly L. Wismath
    Pages 371-388

  12. Iterated Galois Connections in Arithmetic and Linguistics

    • J. Lambek
    Pages 389-397

  13. Deductive Systems and Galois Connections

    • I. Chajda, R. Halaš
    Pages 399-411

  14. A Galois Correspondence for Digital Topology

    • J. Šlapal
    Pages 413-424

  15. Galois Connections in Category Theory, Topology and Logic

    • Werner Gähler
    Pages 425-452

  16. Dyadic Mathematics — Abstractions from Logical Thought

    • R. Wille
    Pages 453-498

  17. Back Matter

    Pages 499-501
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Keywords

About this book

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".

Reviews

From the reviews:

"The book under review is the first one fully dedicated to Galois connections and adjunctions. … I recommend this valuable collection to everybody involved in algebraic research and/or teaching algebra in higher education." (Béla Csákány, Acta Scientiarum Mathematicarum, Vol. 71, 2005)

Editors and Affiliations

  • University of Potsdam, Potsdam, Germany

    K. Denecke

  • University of Hannover, Hannover, Germany

    M. Erné

  • University of Lethbridge, Lethbridge, Canada

    S. L. Wismath

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