Galois Connections and Applications
K. Denecke ¡ M. ErnÊ ¡ S.L. Wismath
āĻ¨āĻā§ ā§¨ā§Ļā§§ā§Š ¡ Mathematics and Its Applications āĻŦāĻ 565 ¡ Springer Science & Business Media
āĻ-āĻŦā§āĻ
502
āĻĒā§āĻˇā§āĻ āĻž
reportāĻ°ā§āĻāĻŋāĻ āĻ āĻ°āĻŋāĻāĻŋāĻ āĻ¯āĻžāĻāĻžāĻ āĻāĻ°āĻž āĻšā§āĻ¨āĻŋ Â āĻāĻ°āĻ āĻāĻžāĻ¨ā§āĻ¨
āĻāĻ āĻ-āĻŦā§āĻā§āĻ° āĻŦāĻŋāĻˇā§ā§
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".
āĻ-āĻŦā§āĻā§ āĻ°ā§āĻāĻŋāĻ āĻĻāĻŋāĻ¨
āĻāĻĒāĻ¨āĻžāĻ° āĻŽāĻ¤āĻžāĻŽāĻ¤ āĻāĻžāĻ¨āĻžāĻ¨āĨ¤
āĻĒāĻ āĻ¨ āĻ¤āĻĨā§āĻ¯
āĻ¸ā§āĻŽāĻžāĻ°ā§āĻāĻĢā§āĻ¨ āĻāĻŦāĻ āĻā§āĻ¯āĻžāĻŦāĻ˛ā§āĻ
Android āĻāĻŦāĻ iPad/iPhone āĻāĻ° āĻāĻ¨ā§āĻ¯ Google Play āĻŦāĻ āĻ
ā§āĻ¯āĻžāĻĒ āĻāĻ¨āĻ¸ā§āĻāĻ˛ āĻāĻ°ā§āĻ¨āĨ¤ āĻāĻāĻŋ āĻāĻĒāĻ¨āĻžāĻ° āĻ
ā§āĻ¯āĻžāĻāĻžāĻāĻ¨ā§āĻā§āĻ° āĻ¸āĻžāĻĨā§ āĻ
āĻā§āĻŽā§āĻāĻŋāĻ āĻ¸āĻŋāĻā§āĻ āĻšā§ āĻ āĻāĻĒāĻ¨āĻŋ āĻ
āĻ¨āĻ˛āĻžāĻāĻ¨ āĻŦāĻž āĻ
āĻĢāĻ˛āĻžāĻāĻ¨ āĻ¯āĻžāĻ āĻĨāĻžāĻā§āĻ¨ āĻ¨āĻž āĻā§āĻ¨ āĻāĻĒāĻ¨āĻžāĻā§ āĻĒā§āĻ¤ā§ āĻĻā§ā§āĨ¤
āĻ˛ā§āĻ¯āĻžāĻĒāĻāĻĒ āĻ āĻāĻŽā§āĻĒāĻŋāĻāĻāĻžāĻ°
Google Play āĻĨā§āĻā§ āĻā§āĻ¨āĻž āĻ
āĻĄāĻŋāĻāĻŦā§āĻ āĻāĻĒāĻ¨āĻŋ āĻāĻŽā§āĻĒāĻŋāĻāĻāĻžāĻ°ā§āĻ° āĻā§ā§āĻŦ āĻŦā§āĻ°āĻžāĻāĻāĻžāĻ°ā§ āĻļā§āĻ¨āĻ¤ā§ āĻĒāĻžāĻ°ā§āĻ¨āĨ¤
eReader āĻāĻŦāĻ āĻ
āĻ¨ā§āĻ¯āĻžāĻ¨ā§āĻ¯ āĻĄāĻŋāĻāĻžāĻāĻ¸
Kobo eReaders-āĻāĻ° āĻŽāĻ¤ā§ e-ink āĻĄāĻŋāĻāĻžāĻāĻ¸ā§ āĻĒāĻĄāĻŧāĻ¤ā§, āĻāĻĒāĻ¨āĻžāĻā§ āĻāĻāĻāĻŋ āĻĢāĻžāĻāĻ˛ āĻĄāĻžāĻāĻ¨āĻ˛ā§āĻĄ āĻ āĻāĻĒāĻ¨āĻžāĻ° āĻĄāĻŋāĻāĻžāĻāĻ¸ā§ āĻā§āĻ°āĻžāĻ¨ā§āĻ¸āĻĢāĻžāĻ° āĻāĻ°āĻ¤ā§ āĻšāĻŦā§āĨ¤ āĻŦā§āĻ¯āĻŦāĻšāĻžāĻ°āĻāĻžāĻ°ā§āĻ° āĻāĻĻā§āĻĻā§āĻļā§āĻ¯ā§ āĻ¤ā§āĻ°āĻŋ āĻ¸āĻšāĻžā§āĻ¤āĻž āĻā§āĻ¨ā§āĻĻā§āĻ°āĻ¤ā§ āĻĻā§āĻā§āĻž āĻ¨āĻŋāĻ°ā§āĻĻā§āĻļāĻžāĻŦāĻ˛ā§ āĻ
āĻ¨ā§āĻ¸āĻ°āĻŖ āĻāĻ°ā§ āĻ¯ā§āĻ¸āĻŦ eReader-āĻ āĻĢāĻžāĻāĻ˛ āĻĒāĻĄāĻŧāĻž āĻ¯āĻžāĻŦā§ āĻ¸ā§āĻāĻžāĻ¨ā§ āĻā§āĻ°āĻžāĻ¨ā§āĻ¸āĻĢāĻžāĻ° āĻāĻ°ā§āĻ¨āĨ¤
