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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 Post's lattice ： ウィキペディア英語版 Post's lattice In logic and universal algebra, Post's lattice denotes the lattice of all clones on a two-element set , ordered by inclusion. It is named for Emil Post, who published a complete description of the lattice in 1941.〔E. L. Post, ''The two-valued iterative systems of mathematical logic'', Annals of Mathematics studies, no. 5, Princeton University Press, Princeton 1941, 122 pp.〕 The relative simplicity of Post's lattice is in stark contrast to the lattice of clones on a three-element (or larger) set, which has the cardinality of the continuum, and a complicated inner structure. A modern exposition of Post's result can be found in Lau (2006).〔D. Lau, ''Function algebras on finite sets: Basic course on many-valued logic and clone theory'', Springer, New York, 2006, 668 pp. ISBN 978-3-540-36022-3〕 ==Basic concepts== A Boolean function, or logical connective, is an ''n''-ary operation for some , where 2 denotes the two-element set . Particular Boolean functions are the projections :$\backslash pi\_k^n(x\_1,\backslash dots,x\_n)=x\_k,$ and given an ''m''-ary function ''f'', and ''n''-ary functions ''g''1, ..., ''g''''m'', we can construct another ''n''-ary function :$h(x\_1,\backslash dots,x\_n)=f(g\_1(x\_1,\backslash dots,x\_n),\backslash dots,g\_m(x\_1,\backslash dots,x\_n)),$ called their composition. A set of functions closed under composition, and containing all projections, is called a clone. Let ''B'' be a set of connectives. The functions which can be defined by a formula using propositional variables and connectives from ''B'' form a clone (), indeed it is the smallest clone which includes ''B''. We call () the clone ''generated'' by ''B'', and say that ''B'' is the ''basis'' of (). For example, (∧ ) are all Boolean functions, and (1, ∧, ∨ ) are the monotone functions. We use the operations ¬ (negation), ∧ (conjunction or meet), ∨ (disjunction or join), → (implication), ↔ (biconditional), + (exclusive disjunction or Boolean ring addition), ↛ (nonimplication), ?: (the ternary conditional operator) and the constant unary functions 0 and 1. Moreover, we need the threshold functions : 0&\text\end For example, th1''n'' is the large disjunction of all the variables ''x''''i'', and th''n''''n'' is the large conjunction. Of particular importance is the majority function :$\backslash mathrm=\backslash mathrm^3\_2=(x\backslash land\; y)\backslash lor(x\backslash land\; z)\backslash lor(y\backslash land\; z).$ We denote elements of 2''n'' (i.e., truth-assignments) as vectors: . The set 2''n'' carries a natural product Boolean algebra structure. That is, ordering, meets, joins, and other operations on ''n''-ary truth assignments are defined pointwise: :$(a\_1,\backslash dots,a\_n)\backslash le(b\_1,\backslash dots,b\_n)\backslash iff\; a\_i\backslash le\; b\_i\backslash texti=1,\backslash dots,n,$ :$(a\_1,\backslash dots,a\_n)\backslash land(b\_1,\backslash dots,b\_n)=(a\_1\backslash land\; b\_1,\backslash dots,a\_n\backslash land\; b\_n).$ 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「Post's lattice」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.