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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 MV-algebra ： ウィキペディア英語版 MV-algebra In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation $\backslash oplus$, a unary operation $\backslash neg$, and the constant $0$, satisfying certain axioms. MV-algebras are the algebraic semantics of Łukasiewicz logic; the letters MV refer to the ''many-valued'' logic of Łukasiewicz. MV-algebras coincide with the class of bounded commutative BCK algebras. ==Definitions== An MV-algebra is an algebraic structure $\backslash langle\; A,\; \backslash oplus,\; \backslash lnot,\; 0\backslash rangle,$ consisting of * a non-empty set $A,$ * a binary operation $\backslash oplus$ on $A,$ * a unary operation $\backslash lnot$ on $A,$ and * a constant $0$ denoting a fixed element of $A,$ which satisfies the following identities: * $(x\; \backslash oplus\; y)\; \backslash oplus\; z\; =\; x\; \backslash oplus\; (y\; \backslash oplus\; z),$ * $x\; \backslash oplus\; 0\; =\; x,$ * $x\; \backslash oplus\; y\; =\; y\; \backslash oplus\; x,$ * $\backslash lnot\; \backslash lnot\; x\; =\; x,$ * $x\; \backslash oplus\; \backslash lnot\; 0\; =\; \backslash lnot\; 0,$ and * $\backslash lnot\; (\; \backslash lnot\; x\; \backslash oplus\; y)\backslash oplus\; y\; =\; \backslash lnot\; (\; \backslash lnot\; y\; \backslash oplus\; x)\; \backslash oplus\; x.$ By virtue of the first three axioms, $\backslash langle\; A,\; \backslash oplus,\; 0\; \backslash rangle$ is a commutative monoid. Being defined by identities, MV-algebras form a variety of algebras. The variety of MV-algebras is a subvariety of the variety of BL-algebras and contains all Boolean algebras. An MV-algebra can equivalently be defined (Hájek 1998) as a prelinear commutative bounded integral residuated lattice $\backslash langle\; L,\; \backslash wedge,\; \backslash vee,\; \backslash otimes,\; \backslash rightarrow,\; 0,\; 1\; \backslash rangle$ satisfying the additional identity $x\; \backslash vee\; y\; =\; (x\; \backslash rightarrow\; y)\; \backslash rightarrow\; y.$ 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「MV-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.