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 $$ X^\Delta = \{ a \in M : a \ge x \ \text{for all}\ x \in X \} $$ $$ X^\nabla = \{ a \in M : a \le x \ \text{for all}\ x \in X \} $$ The condition $\phi(X) = (X^\Delta)^\nabla$ defines a closure operation (cf. Closure relation) $\phi$ on $\mathcal{P}(M)$. The lattice $L$ of all $\phi$-closed subsets of $\mathcal{P}(M)$ is complete. For any $x \in M$ the set $(x^\Delta)^\nabla$ is the principal ideal generated by $x$. Put $i(x) = (x^\Delta)^\nabla$ 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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