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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 Anticommutativity ： ウィキペディア英語版 Anticommutativity In mathematics, anticommutativity is the property of an operation with two or more arguments wherein swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric operations. == Definition == An $n$-ary operation is anticommutative if swapping the order of any two arguments negates the result. For example, a binary operation ∗ is anti-commutative if for all ''x'' and ''y'', :. More formally, a map $\backslash scriptstyle$ *:A^n \to \mathfrak from the set of all ''n''-tuples of elements in a set ''A'' (where ''n'' is a general integer) to a group $\backslash scriptstyle\backslash mathfrak$ is anticommutative if and only if :$=\; \backslash sgn(\backslash sigma)\; ($ *\dots * x_}) \qquad \forall\boldsymbol = (x_1,x_2,\dots,x_n) \in A^n where $\backslash scriptstyle\backslash sigma:(n)\backslash to(n)$ is an arbitrary permutation of the set (''n'') of the first ''n'' positive integers and $\backslash mathrm(\backslash sigma)$ is its sign. This equality expresses the following concept: * the value of the operation is unchanged, when applied to all ordered tuples constructed by even permutation of the elements of a fixed one. * the value of the operation is the inverse of its value on a fixed tuple, when applied to all ordered tuples constructed by odd permutation to the elements of the fixed one. The need for the existence of this inverse element is the main reason for requiring the codomain $\backslash scriptstyle\backslash mathfrak$ of the operation to be at least a group. Note that this is an abuse of notation, since the codomain of the operation needs only to be a group: "−1" does not have a precise meaning since a multiplication is not necessarily defined on $\backslash scriptstyle\backslash mathfrak$. Particularly important is the case . A binary operation $\backslash scriptstyle$ *:A\times A\to \mathfrak is anticommutative if and only if :$x\_1$ * x_2 = -(x_2 * x_1) \qquad\forall(x_1,x_2)\in A\times A This means that is the inverse of the element in $\backslash scriptstyle\backslash mathfrak$. 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「Anticommutativity」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.