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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Mirsky's theorem ： ウィキペディア英語版 Mirsky's theorem In mathematics, in the areas of order theory and combinatorics, Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the order into a minimum number of antichains. It is named for and is closely related to Dilworth's theorem on the widths of partial orders, to the perfection of comparability graphs, to the Gallai–Hasse–Roy–Vitaver theorem relating longest paths and colorings in graphs, and to the Erdős–Szekeres theorem on monotonic subsequences. ==The theorem== The height of a partially ordered set is defined to be the maximum cardinality of a chain, a totally ordered subset of the given partial order. For instance, in the set of positive integers from 1 to ''N'', ordered by divisibility, one of the largest chains consists of the powers of two that lie within that range, from which it follows that the height of this partial order is $1+\backslash lfloor\backslash log\_2\; N\backslash rfloor$. Mirsky's theorem states that, for every partially ordered set, the height also equals the minimum number of antichains (subsets in which no pair of elements are ordered) into which the set may be partitioned. In such a partition, every two elements of the longest chain must go into two different antichains, so the number of antichains is always greater than or equal to the height; another formulation of Mirsky's theorem is that there always exists a partition for which the number of antichains equals the height. Again, in the example of positive integers ordered by divisibility, the numbers can be partitioned into the antichains , , , etc. There are $1+\backslash lfloor\backslash log\_2\; N\backslash rfloor$ sets in this partition, and within each of these sets, every pair of numbers forms a ratio less than two, so no two numbers within one of these sets can be divisible. To prove the existence of a partition into a small number of antichains for an arbitrary finite partially ordered set, consider for every element ''x'' the chains that have ''x'' as their largest element, and let ''N''(''x'') denote the size of the largest of these ''x''-maximal chains. Then each set ''N''−1(''i''), consisting of elements that have equal values of ''N'', is an antichain, and these antichains partition the partial order into a number of antichains equal to the size of the largest chain. In his original proof, Mirsky constructs the same partition inductively, by choosing an antichain of the maximal elements of longest chains, and showing that the length of the longest chain among the remaining elements is reduced by one. 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「Mirsky's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.