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Galois Connections and Applications : Mathematics and Its Applications - Klaus Denecke

Galois Connections and Applications

Mathematics and Its Applications

By: Klaus Denecke (Editor), S. L. Wismath (Editor), M. Erne (Editor)

Hardcover

Published: 31st March 2004
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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".

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." (Bela Csakany, Acta Scientiarum Mathematicarum, Vol. 71, 2005)

Preface
Adjunctions and Galois Connections: Origins, History and Development
Categorical Galois Theory: Revision and Some Recent Developments
The Polarity between Approximation and Distribution
Galois Connections and Complete Sublattices
Galois Connections for Operations and Relations
Galois Connections and Polynomial Completeness
Q-Independence and Weak Automorphisms
A Survey of Clones Closed Under Conjugation
Galois Connections for Partial Algebras
Complexity of Terms and the Galois Connection Id-Mod
Iterated Galois Connections in Arithmetic and Linguistics
Deductive Systems and Galois Connections
A Galois Correspondence for Digital Topology
Galois Connections in Category Theory
Dyadic Mathematics - Abstractions from Logical Thought
Index"
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9781402018978
ISBN-10: 1402018975
Series: Mathematics and Its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 502
Published: 31st March 2004
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 1.99