Mathematics and Its Applications

Galois Connections and Applications

Editors: Denecke, Klaus, Erné, M., Wismath, S.L. (Eds.)

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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)


Table of contents (15 chapters)

  • Adjunctions and Galois Connections: Origins, History and Development

    Erné, Marcel

    Pages 1-138

  • Categorical Galois Theory: Revision and Some Recent Developments

    Janelidze, G.

    Pages 139-171

  • The Polarity between Approximation and Distribution

    Erné, Marcel

    Pages 173-210

  • Galois Connections and Complete Sublattices

    Denecke, Klaus (et al.)

    Pages 211-229

  • Galois Connections for Operations and Relations

    Pöschel, Reinhard

    Pages 231-258

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-1-4020-1898-5
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.00
price for USA
  • ISBN 978-1-4020-1897-8
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $139.00
price for USA
  • ISBN 978-90-481-6540-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Galois Connections and Applications
Editors
  • Klaus Denecke
  • M. Erné
  • S.L. Wismath
Series Title
Mathematics and Its Applications
Series Volume
565
Copyright
2004
Publisher
Springer Netherlands
Copyright Holder
Springer Science+Business Media Dordrecht
eBook ISBN
978-1-4020-1898-5
DOI
10.1007/978-1-4020-1898-5
Hardcover ISBN
978-1-4020-1897-8
Softcover ISBN
978-90-481-6540-7
Edition Number
1
Number of Pages
XVI, 502
Topics