View source for Completion, MacNeille (of a partially ordered set)

''completion by sections'', ''Dedekind–MacNeille completion''

The [[Complete lattice|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|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.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  H.M. MacNeille,   "Partially ordered sets"  ''Trans. Amer. Math. Soc.'' , '''42'''  (1937)  pp. 416–460</TD></TR>
</table>



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

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S.P. Crawley,   "Regular embeddings which preserve lattice structure"  ''Proc. Amer. Math. Soc.'' , '''13'''  (1962)  pp. 748–752</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.T. Johnstone,   "Stone spaces" , Cambridge Univ. Press  (1982)</TD></TR>
</table>

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[[Category:Order, lattices, ordered algebraic structures]]