Galois Connections and Applications
(Sprache: Englisch)
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....
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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".
Inhaltsverzeichnis zu „Galois Connections and Applications “
PrefaceM. Erné; Adjunctions and Galois Connections: Origins, History and DevelopmentG. Janelidze; Categorical Galois Theory: Revision and Some Recent DevelopmentsM. Erné; The Polarity between Approximation and DistributionK. Denecke, S.L. Wismath; Galois Connections and Complete SublatticesR. Pöschel; Galois Connections for Operations and RelationsK. Kaarli; Galois Connections and Polynomial CompletenessK. Glazek, St. Niwczyk; Q-Independence and Weak AutomorphismsA. Szendrei; A Survey of Clones Closed Under ConjugationP. Burmeister; Galois Connections for Partial AlgebrasK. Denecke, S.L. Wismath; Complexity of Terms and the Galois Connection Id-ModJ. Lambek; Iterated Galois Connections in Arithmetic and LinguisticsI. Chajda, R. Halas; Deductive Systems and Galois ConnectionsJ. Slapal; A Galois Correspondence for Digital TopologyW. Gähler; Galois Connections in Category Theory, Topology and LogicR. Wille; Dyadic Mathematics - Abstractions from Logical ThoughtIndex
Bibliographische Angaben
- 2010, 502 Seiten, Maße: 23,5 cm, Kartoniert (TB), Englisch
- Herausgegeben:Denecke, K.; Erné, M.; Wismath, S.L.
- Verlag: Springer Netherlands
- ISBN-10: 9048165407
- ISBN-13: 9789048165407
Sprache:
Englisch
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