Completion, MacNeille (of a partially ordered set)
completion by sections
The complete lattice obtained from a partially ordered set
in the following way. Let
be the set of all subsets of
, ordered by inclusion. For any
assume that
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The condition defines a closure operation (cf. Closure relation)
on
. The lattice
of all
-closed subsets of
is complete. For any
the set
is the principal ideal generated by
. Put
for all
. Then
is an isomorphic imbedding of
into the complete lattice
that preserves all least upper bounds and greatest lower bounds existing in
. 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
[1] | H.M. MacNeille, "Partially ordered sets" Trans. Amer. Math. Soc. , 42 (1937) pp. 416–460 |
Comments
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).
References
[a1] | S.P. Crawley, "Regular embeddings which preserve lattice structure" Proc. Amer. Math. Soc. , 13 (1962) pp. 748–752 |
[a2] | P.T. Johnstone, "Stone spaces" , Cambridge Univ. Press (1982) |
Completion, MacNeille (of a partially ordered set). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completion,_MacNeille_(of_a_partially_ordered_set)&oldid=33797