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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 Commutative non-associative magmas ： ウィキペディア英語版 Commutative non-associative magmas In mathematics, there exist magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras. == A commutative non-associative magma derived from the rock, paper, scissors game == Let $M\; :=\; \backslash$ , standing for the "rock", "scissors" and "paper" gestures respectively, and consider the binary operation $\backslash cdot\; :\; M\; \backslash times\; M\; \backslash to\; M$ derived from the rules of the game as follows: : For all $x,\; y\; \backslash in\; M$: : * If $x\; \backslash neq\; y$ and $x$ beats $y$ in the game, then $x\; \backslash cdot\; y\; =\; y\; \backslash cdot\; x\; =\; x$ : * $x\; \backslash cdot\; x\; =\; x$     I.e. every $x$ is idempotent. : So that for example: : * $r\; \backslash cdot\; p\; =\; p\; \backslash cdot\; r\; =\; p$   "paper beats rock"; : * $s\; \backslash cdot\; s\; =\; s$   "scissors tie with scissors". This results in the Cayley table: $\backslash begin$ \cdot & r & p & s\\ \hline r & r & p & r\\ p & p & p & s\\ s & r & s & s \end By definition, the magma $(M,\; \backslash cdot)$ is commutative, but it is also non-associative, as shown by: :$r\; \backslash cdot\; (p\; \backslash cdot\; s)\; =\; r\; \backslash cdot\; s\; =\; r\; \backslash neq\; s\; =\; p\; \backslash cdot\; s\; =\; (r\; \backslash cdot\; p)\; \backslash cdot\; s$ i.e. :$r\; \backslash cdot\; (p\; \backslash cdot\; s)\; \backslash neq\; (r\; \backslash cdot\; p)\; \backslash cdot\; s$ 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「Commutative non-associative magmas」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.