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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 Transversal (combinatorics) ： ウィキペディア英語版 Transversal (combinatorics) In mathematics, given a collection ''C'' of sets, a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of ''C'' (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal. One variation, the one that mimics the situation when the sets are mutually disjoint, is that there is a bijection ''f'' from the transversal to ''C'' such that ''x'' is an element of ''f''(''x'') for each ''x'' in the transversal. In this case, the transversal is also called a system of distinct representatives. The other, less commonly used, possibility does not require a one-to-one relation between the elements of the transversal and the sets of ''C''. Loosely speaking, in this situation the members of the ''system of representatives'' are not necessarily distinct. ==Examples== In group theory, given a subgroup ''H'' of a group ''G'', a right (respectively left) transversal is a set containing exactly one element from each right (respectively left) coset of ''H''. In this case, the "sets" (cosets) are mutually disjoint, i.e. the cosets form a partition of the group. As a particular case of the previous example, given a direct product of groups $G\; =\; H\; \backslash times\; K$, then ''H'' is a transversal for the cosets of ''K''. In general, since any equivalence relation on an arbitrary set gives rise to a partition, picking any representative from each equivalence class results in a transversal. Another instance of a partition-based transversal occurs when one considers the equivalence relation known as the (set-theoretic) kernel of a function, defined for a function $f$ with domain ''X'' as the partition of the domain $\backslash operatorname\; f\; :=\; \backslash left\backslash$. which partitions the domain of ''f'' into equivalence classes such that all elements in a class map via ''f'' to the same value. If ''f'' is injective, there is only one transversal of $\backslash operatorname\; f$. For a not-necessarily-injective ''f'', fixing a transversal ''T'' of $\backslash operatorname\; f$ induces a one-to-one correspondence between ''T'' and the image of ''f'', henceforth denoted by $\backslash operatornamef$. Consequently, a function $g:\; (\backslash operatorname\; f)\; \backslash to\; T$ is well defined by the property that for all ''z'' in $\backslash operatorname\; f,\; g(z)=x$ where ''x'' is the unique element in ''T'' such that $f(x)=z$; furthermore, ''g'' can be extended (not necessarily in a unique manner) so that it is defined on the whole codomain of ''f'' by picking arbitrary values for ''g(z)'' when ''z'' is outside the image of ''f''. It is a simple calculation to verify that ''g'' thus defined has the property that $f\backslash circ\; g\; \backslash circ\; f\; =\; f$, which is the proof (when the domain and codomain of ''f'' are the same set) that the full transformation semigroup is a regular semigroup. $g$ acts as a (not necessarily unique) quasi-inverse for ''f''; within semigroup theory this is simply called an inverse. Note however that for an arbitrary ''g'' with the aforementioned property the "dual" equation $g\; \backslash circ\; f\; \backslash circ\; g=\; g$ may not hold. However if we denote by $h=\; g\; \backslash circ\; f\; \backslash circ\; g$, then ''f'' is a quasi-inverse of ''h'', i.e. $h\; \backslash circ\; f\; \backslash circ\; h\; =\; h$. Hall's marriage theorem gives necessary and sufficient conditions for a finite collection of not necessarily distinct, but non-empty sets to have a transversal. 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「Transversal (combinatorics)」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.