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''completion by sections''
 
''completion by sections''
  
The [[Complete lattice|complete lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c0240701.png" /> obtained from a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c0240702.png" /> in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c0240703.png" /> be the set of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c0240704.png" />, ordered by inclusion. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c0240705.png" /> assume that
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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
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$$
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X^\Delta = \{ a \in M : a \ge x \ \text{for all}\ x \in X \}
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$$
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$$
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X^\nabla = \{ a \in M : a \le x \ \text{for all}\ x \in X \}
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$$
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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 $M$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c0240706.png" /></td> </tr></table>
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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.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c0240707.png" /></td> </tr></table>
 
 
 
The condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c0240708.png" /> defines a closure operation (cf. [[Closure relation|Closure relation]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c0240709.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407010.png" />. The lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407011.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407012.png" />-closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407013.png" /> is complete. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407014.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407015.png" /> is the principal ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407016.png" />. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407018.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407019.png" /> is an isomorphic imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407020.png" /> into the complete lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407021.png" /> that preserves all least upper bounds and greatest lower bounds existing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024070/c02407022.png" />. 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====
 
====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>
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<table>
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<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>
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</table>
  
  
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====References====
 
====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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<table>
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<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>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.T. Johnstone,  "Stone spaces" , Cambridge Univ. Press  (1982)</TD></TR>
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</table>
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{{TEX|done}}

Revision as of 14:28, 18 October 2014

completion by sections

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.

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)
How to Cite This Entry:
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=17175
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article