[過去ﾛｸﾞ] 現代数学の系譜　カントル　超限集合論 (1002ﾚｽ)

現代数学の系譜　カントル　超限集合論 http://rio2016.5ch.net/test/read.cgi/math/1570237031/

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323: 現代数学の系譜 雑談 ◆e.a0E5TtKE [] 2019/10/13(日) 07:22:41.21 ID:sXrN/kYa >>322 参考追加 https://en.wikipedia.org/wiki/Order_theory Order theory (抜粋) Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Contents 1 Background and motivation 2 Basic definitions 2.1 Partially ordered sets 2.2 Visualizing a poset 2.3 Special elements within an order 2.4 Duality 2.5 Constructing new orders 3 Functions between orders 4 Special types of orders 5 Subsets of ordered sets 6 Related mathematical areas 6.1 Universal algebra 6.2 Topology 6.3 Category theory 7 History 8 See also つづく http://rio2016.5ch.net/test/read.cgi/math/1570237031/323

324: 現代数学の系譜 雑談 ◆e.a0E5TtKE [] 2019/10/13(日) 07:23:50.76 ID:sXrN/kYa >>323 つづき Category theory The visualization of orders with Hasse diagrams has a straightforward generalization: instead of displaying lesser elements below greater ones, the direction of the order can also be depicted by giving directions to the edges of a graph. In this way, each order is seen to be equivalent to a directed acyclic graph, where the nodes are the elements of the poset and there is a directed path from a to b if and only if a ? b. Dropping the requirement of being acyclic, one can also obtain all preorders. When equipped with all transitive edges, these graphs in turn are just special categories, where elements are objects and each set of morphisms between two elements is at most singleton. Functions between orders become functors between categories. Many ideas of order theory are just concepts of category theory in small. For example, an infimum is just a categorical product. More generally, one can capture infima and suprema under the abstract notion of a categorical limit (or colimit, respectively). Another place where categorical ideas occur is the concept of a (monotone) Galois connection, which is just the same as a pair of adjoint functors. But category theory also has its impact on order theory on a larger scale. Classes of posets with appropriate functions as discussed above form interesting categories. Often one can also state constructions of orders, like the product order, in terms of categories. Further insights result when categories of orders are found categorically equivalent to other categories, for example of topological spaces. This line of research leads to various representation theorems, often collected under the label of Stone duality. つづく http://rio2016.5ch.net/test/read.cgi/math/1570237031/324

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