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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 Iterated binary operation ： ウィキペディア英語版 Iterated binary operation In mathematics, an iterated binary operation is an extension of a binary operation on a set ''S'' to a function on finite sequences of elements of ''S'' through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the multiplication operation to the product operation. Other operations, e.g., the set theoretic operations union and intersection, are also often iterated, but the iterations are not given separate names. In print, summation and product are represented by special symbols; but other iterated operators often are denoted by larger variants of the symbol for the ordinary binary operator. Thus, the iterations of the four operations mentioned above are denoted ::$\backslash sum,\backslash \; \backslash prod,\backslash \; \backslash bigcup,$ and $\backslash bigcap$, respectively. More generally, iteration of a binary function is generally denoted by a slash: iteration of $f$ over the sequence $(a\_,\; a\_\; \backslash ldots,\; a\_)$ is denoted by $f\; /\; (a\_,\; a\_\; \backslash ldots,\; a\_)$. In general, there is more than one way to extend a binary operation to operate on finite sequences, depending on whether the operator is associative, and whether the operator has identity elements. == Definition == Denote by a''j'',''k'', with and , the finite sequence of length of elements of ''S'', with members (''a''i), for . Note that if , the sequence is empty. For , define a new function ''F''''l'' on finite nonempty sequences of elements of ''S'', where :$F\_l(\backslash mathbf\_)=$ \begin a_0, &k=1\\ f(F_l(\mathbf_), a_), &k>1 \end. Similarly, define :$F\_r(\backslash mathbf\_)=$ \begin a_0, &k=1\\ f(a_0, F_r(\mathbf_)), &k>1 \end. If ''f'' has a unique left identity ''e'', the definition of ''F''''l'' can be modified to operate on empty sequences by defining the value of ''F''''l'' on an empty sequence to be ''e'' (the previous base case on sequences of length 1 becomes redundant). Similarly, ''F''''r'' can be modified to operate on empty sequences if ''f'' has a unique right identity. If ''f'' is associative, then ''F''''l'' equals ''F''''r'', and we can simply write ''F''. Moreover, if an identity element ''e'' exists, then it is unique (see Monoid). If ''f'' is commutative and associative, then ''F'' can operate on any non-empty finite multiset by applying it to an arbitrary enumeration of the multiset. If ''f'' moreover has an identity element ''e'', then this is defined to be the value of ''F'' on an empty multiset. If ''f'' is idempotent, then the above definitions can be extended to finite sets. If ''S'' also is equipped with a metric or more generally with topology that is Hausdorff, so that the concept of a limit of a sequence is defined in ''S'', then an ''infinite iteration'' on a countable sequence in ''S'' is defined exactly when the corresponding sequence of finite iterations converges. Thus, e.g., if ''a0'', ''a1'', ''a2'', ''a3'', ... is an infinite sequence of real numbers, then the infinite product $\backslash prod\_^\backslash infty\; a\_i\backslash ,$ is defined, and equal to $\backslash lim\backslash limits\_\backslash prod\_^na\_i,$  if and only if that limit exists. 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「Iterated binary operation」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.