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Dedekind–MacNeille completion
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically <a href="/facts/Order_theory/gn6fM3oJ">order theory</a>, the Dedekind–MacNeille completion of a <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> is the smallest <a href="/facts/Complete_lattice/pQbphWeR">complete lattice</a> that contains it. It is named after <a href="/facts/Holbrook_Mann_MacNeille/nPNSje4q">Holbrook Mann MacNeille</a> whose 1937 paper first defined and constructed it, and after <a href="/facts/Richard_Dedekind/CLkWDhY3">Richard Dedekind</a> because its construction generalizes the <a href="/facts/Dedekind_cut/rqhbb24J">Dedekind cuts</a> used by Dedekind to construct the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> from the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>. It is also called the completion by cuts or normal completion. </p>