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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 Matroid rank ： ウィキペディア英語版 Matroid rank In the mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset ''S'' of elements of the matroid is, similarly, the maximum size of an independent subset of ''S'', and the rank function of the matroid maps sets of elements to their ranks. The rank function is one of the fundamental concepts of matroid theory via which matroids may be axiomatized. The rank functions of matroids form an important subclass of the submodular set functions, and the rank functions of the matroids defined from certain other types of mathematical object such as undirected graphs, matrices, and field extensions are important within the study of those objects. ==Properties and axiomatization== The rank function of a matroid obeys the following properties. *The value of the rank function is always a non-negative integer. *For any two subsets $A$ and $B$ of $E$, $r(A\backslash cup\; B)+r(A\backslash cap\; B)\backslash le\; r(A)+r(B)$. That is, the rank is a submodular function. *For any set $A$ and element $x$, $r(A)\backslash le\; r(A\backslash cup\backslash )\backslash le\; r(A)+1$. From the first of these two inequalities it follows more generally that, if $A\backslash subset\; B\backslash subset\; E$, then $r(A)\backslash leq\; r(B)\backslash leq\; r(E)$. That is, the rank is a monotonic function. These properties may be used as axioms to characterize the rank function of matroids: every integer-valued submodular function on the subsets of a finite set that obeys the inequalities $r(A)\backslash le\; r(A\backslash cup\backslash )\backslash le\; r(A)+1$ for all $A$ and $x$ is the rank function of a matroid.〔.〕〔.〕 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「Matroid rank」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.