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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 Relation composition ： ウィキペディア英語版 Composition of relations In mathematics, the composition of binary relations is a concept of forming a new relation from two given relations ''R'' and ''S'', having as its most well-known special case the composition of functions. == Definition == If $R\backslash subseteq\; X\backslash times\; Y$ and $S\backslash subseteq\; Y\backslash times\; Z$ are two binary relations, then their composition $S\backslash circ\; R$ is the relation :$S\backslash circ\; R\; =\; \backslash .$ In other words, $S\backslash circ\; R\backslash subseteq\; X\backslash times\; Z$ is defined by the rule that says $(x,z)\backslash in\; S\backslash circ\; R$ if and only if there is an element $y\backslash in\; Y$ such that $x\backslash ,R\backslash ,y\backslash ,S\backslash ,z$ (i.e. $(x,y)\backslash in\; R$ and $(y,z)\backslash in\; S$). In particular fields, authors might denote by what is defined here to be . The convention chosen here is such that function composition (with the usual notation) is obtained as a special case, when ''R'' and ''S'' are functional relations. Some authors〔Kilp, Knauer & Mikhalev, p. 7〕 prefer to write $\backslash circ\_l$ and $\backslash circ\_r$ explicitly when necessary, depending whether the left or the right relation is the first one applied. A further variation encountered in computer science is the Z notation: $\backslash circ$ is used to denote the traditional (right) composition, but ⨾ ; (a fat open semicolon with Unicode code point U+2A3E) denotes left composition.〔ISO/IEC 13568:2002(E), p. 23〕〔http://www.fileformat.info/info/unicode/char/2a3e/index.htm〕 This use of semicolon coincides with the notation for function composition used (mostly by computer scientists) in Category theory,〔http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf, p. 6〕 as well as the notation for dynamic conjunction within linguistic dynamic semantics.〔http://plato.stanford.edu/entries/dynamic-semantics/#EncDynTypLog〕 The semicolon notation (with this semantic) was introduced by Ernst Schröder in 1895.〔 A free HTML version of the book is available at http://www.cs.man.ac.uk/~pt/Practical_Foundations/〕 The binary relations $R\backslash subseteq\; X\backslash times\; Y$ are sometimes regarded as the morphisms $R\backslash colon\; X\backslash to\; Y$ in a category Rel which has the sets as objects. In Rel, composition of morphisms is exactly composition of relations as defined above. The category Set of sets is a subcategory of Rel that has the same objects but fewer morphisms. A generalization of this is found in the theory of allegories. 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「Composition of relations」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.