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In the mathematical field of order theory, a containment order is the partial order that arises as the subset-containment relation on some collection of objects. In a simple way, every poset ''P'' = (≤, ''X'') is (isomorphic to) a containment order (just as every group is isomorphic to a permutation group - see Cayley's theorem). To see this, associate to each element ''x'' of ''X'' the set : then the transitivity of ≤ ensures that for all ''a'' and ''b'' in ''X'', we have : There can be sets of cardinality less than such that ''P'' is isomorphic to the containment order on S. The size of the smallest possible ''S'' is called the 2-dimension of ''S''. Several important classes of poset arise as containment orders for some natural collections, like the Boolean lattice ''Qn'', which is the collection of all 2''n'' subsets of an ''n''-element set, the interval-containment orders, which are precisely the orders of order dimension at most two, and the dimension-''n'' orders, which are the containment orders on collections of ''n''-boxes anchored at the origin. Other containment orders that are interesting in their own right include the circle orders, which arise from disks in the plane, and the angle orders. == See also == *Birkhoff's representation theorem *Intersection graph *Interval order 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「containment order」の詳細全文を読む スポンサード リンク
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