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  • Open Daily: 10am - 10pm
    Alley-side Pickup: 10am - 7pm

    3038 Hennepin Ave Minneapolis, MN
    612-822-4611

Open Daily: 10am - 10pm | Alley-side Pickup: 10am - 7pm
3038 Hennepin Ave Minneapolis, MN
612-822-4611
Galois Connections and Applications

Galois Connections and Applications

Denecke, K.
Erné, M.
Wismath, S. L.

Hardcover

Series: Mathematics and Its Applications, Book 565

AlgebraGeneral Mathematics

ISBN10: 1402018975
ISBN13: 9781402018978
Publisher: Springer Nature
Published: Mar 31 2004
Pages: 502
Weight: 2.31
Height: 1.04 Width: 6.26 Depth: 9.66
Language: English
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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