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: ''"Dedekind completion" redirects here. For some other related concepts, see Dedekind completeness.'' In order-theoretic mathematics, the Dedekind–MacNeille completion of a partially ordered set (also called the completion by cuts or normal completion)〔; .〕 is the smallest complete lattice that contains the given partial order. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and constructed it, and after Richard Dedekind because its construction generalizes the Dedekind cuts used by Dedekind to construct the real numbers from the rational numbers. ==Order embeddings and lattice completions== A partially ordered set consists of a set of elements together with a binary relation on pairs of elements that is reflexive ( for every ''x''), transitive (if and then ), and antisymmetric (if both and hold, then ). The usual numeric orderings on the integers or real numbers satisfy these properties; however, unlike the orderings on the numbers, a partial order may have two elements that are ''incomparable'': neither nor holds. Another familiar example of a partial ordering is the inclusion ordering ⊆ on pairs of sets. If is a partially ordered set, a ''completion'' of means a complete lattice with an order-embedding of into .〔, definition 5.3.1, p. 119.〕 The notion of a complete lattice means that every subset of elements of has a unique least upper bound and a unique greatest lower bound; this generalizes the analogous upper bound and lower bound properties of the real numbers. The notion of an order-embedding enforces the requirements that distinct elements of must be mapped to distinct elements of , and that each pair of elements in has the same ordering in as they do in . The real numbers (together with +∞ and −∞) are a completion in this sense of the rational numbers: the set of rational numbers does not have a rational least upper bound, but in the real numbers it has the least upper bound . A given partially ordered set may have several different completions. For instance, one completion of any partially ordered set is the set of its downwardly closed subsets ordered by inclusion. is embedded in this lattice by mapping each element to the lower set of elements that are less than or equal to . The result is a distributive lattice and is used in Birkhoff's representation theorem. However, it may have many more elements than are needed to form a completion of .〔.〕 Among all possible lattice completions, the Dedekind–MacNeille completion is the smallest complete lattice with embedded in it.〔 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「Dedekind–MacNeille completion」の詳細全文を読む スポンサード リンク
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