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In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets. Formally, a complete lattice ''L'' is said to be completely distributive if, for any doubly indexed family of ''L'', we have : where ''F'' is the set of choice functions ''f'' choosing for each index ''j'' of ''J'' some index ''f''(''j'') in ''K''''j''.〔B. A. Davey and H. A. Priestey, ''Introduction to Lattices and Order'' 2nd Edition, Cambridge University Press, 2002, ISBN 0-521-78451-4〕 Complete distributivity is a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices.〔 Without the axiom of choice, no complete lattice with more than one element can ever satisfy the above property, as one can just let ''x''''j'',''k'' equal the top element of ''L'' for all indices ''j'' and ''k'' with all of the sets ''K''''j'' being nonempty but having no choice function. ==Alternative characterizations== Various different characterizations exist. For example, the following is an equivalent law that avoids the use of choice functions. For any set ''S'' of sets, we define the set ''S''# to be the set of all subsets ''X'' of the complete lattice that have non-empty intersection with all members of ''S''. We then can define complete distributivity via the statement : The operator ( )# might be called the crosscut operator. This version of complete distributivity only implies the original notion when admitting the Axiom of Choice. 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「Completely distributive lattice」の詳細全文を読む スポンサード リンク
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