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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 Robbins algebra ： ウィキペディア英語版 Robbins algebra In abstract algebra, a Robbins algebra is an algebra containing a single binary operation, usually denoted by $\backslash lor$, and a single unary operation usually denoted by $\backslash neg$. These operations satisfy the following axioms: For all elements ''a'', ''b'', and ''c'': # Associativity: $a\; \backslash lor\; \backslash left(b\; \backslash lor\; c\backslash right)\; =\; \backslash left(a\; \backslash lor\; b\backslash right)\; \backslash lor\; c$ # Commutativity: $a\; \backslash lor\; b\; =\; b\; \backslash lor\; a$ # ''Robbins equation'': $\backslash neg\; \backslash left(\; \backslash neg\; \backslash left(a\; \backslash lor\; b\; \backslash right)\; \backslash lor\; \backslash neg\; \backslash left(a\; \backslash lor\; \backslash neg\; b\; \backslash right)\; \backslash right)\; =\; a$ For many years, it was conjectured, but unproven, that all Robbins algebras are Boolean algebras. This was proved in 1996, so the term "Robbins algebra" is now simply a synonym for "Boolean algebra". == History == In 1933, Edward Huntington proposed a new set of axioms for Boolean algebras, consisting of (1) and (2) above, plus: *''Huntingtons equation'': $\backslash neg(\backslash neg\; a\; \backslash lor\; b)\; \backslash lor\; \backslash neg(\backslash neg\; a\; \backslash lor\; \backslash neg\; b)\; =\; a.$ From these axioms, Huntington derived the usual axioms of Boolean algebra. Very soon thereafter, Herbert Robbins posed the "Robbins conjecture", namely that the Huntington equation could be replaced with what came to be called the Robbins equation, and the result would still be Boolean algebra. $\backslash lor$ would interpret Boolean join and $\backslash neg$ Boolean complement. Boolean meet and the constants 0 and 1 are easily defined from the Robbins algebra primitives. Pending verification of the conjecture, the system of Robbins was called "Robbins algebra." Verifying the Robbins conjecture required proving Huntingtons equation, or some other axiomatization of a Boolean algebra, as theorems of a Robbins algebra. Huntington, Robbins, Alfred Tarski, and others worked on the problem, but failed to find a proof or counterexample. William McCune proved the conjecture in 1996, using the automated theorem prover EQP. For a complete proof of the Robbins conjecture in one consistent notation and following McCune closely, see Mann (2003). Dahn (1998) simplified McCune's machine proof. 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「Robbins algebra」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.