Completion, MacNeille (of a partially ordered set)

completion by sections, Dedekind–MacNeille completion

The complete lattice $L$ obtained from a partially ordered set $M$ in the following way. Let $\mathcal{P}(M)$ be the set of all subsets of $M$, ordered by inclusion. For any $X\in \mathcal{P}(M)$ assume that $$X^{\mathrm{\Delta }}=\{a\in M:a\ge x\text{for all}x\in X\}$$ $$X^{\mathrm{\nabla }}=\{a\in M:a\le x\text{for all}x\in X\}$$ The condition $\varphi (X)=(X^{\mathrm{\Delta }}{)}^{\mathrm{\nabla }}$ defines a closure operation (cf. Closure relation) $\varphi$ on $\mathcal{P}(M)$. The lattice $L$ of all $\varphi$-closed subsets of $\mathcal{P}(M)$ is complete. For any $x\in M$ the set $(x^{\mathrm{\Delta }}{)}^{\mathrm{\nabla }}$ is the principal ideal generated by $x$. Put $i(x)=(x^{\mathrm{\Delta }}{)}^{\mathrm{\nabla }}$ for all $x\in M$. Then $i$ is an isomorphic imbedding of $M$ into the complete lattice $L$ that preserves all least upper bounds and greatest lower bounds existing in $M$.

When applied to the ordered set of rational numbers, the construction described above gives the completion of the set of rational numbers by Dedekind sections.

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The MacNeille completion of a Boolean algebra is a (complete) Boolean algebra, but the MacNeille completion of a distributive lattice need not be distributive (see [a1]). When restricted to Boolean algebras the MacNeille completion corresponds by Stone duality (cf. Stone space) to the construction of the absolute (or the Gleason cover construction) for compact zero-dimensional spaces (cf. Zero-dimensional space; [a2], p. 109).

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