Dedekind–MacNeille completion

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.

A partially ordered set consists of a set of elements together with a binary relation x ≤ y on pairs of elements that is reflexive (x ≤ x for every x), transitive (if x ≤ y and y ≤ z then x ≤ z), and antisymmetric (if both x ≤ y and y ≤ x hold, then x = y). 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 x ≤ y nor y ≤ x holds. Another familiar example of a partial ordering is the inclusion ordering ⊆ on pairs of sets.

If S is a partially ordered set, a completion of S means a complete lattice L with an order-embedding of S into L. The notion of a complete lattice means that every subset of elements of L has infimum and supremum; this generalizes the analogous properties of the real numbers. The notion of an order-embedding enforces the requirements that distinct elements of S must be mapped to distinct elements of L, and that each pair of elements in S has the same ordering in L as they do in S. The extended real number line (real numbers together with +∞ and −∞) is a completion in this sense of the rational numbers: the set of rational numbers {3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...} does not have a rational least upper bound, but in the real numbers it has the least upper bound π.

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