Difference between revisions of "Completion, MacNeille (of a partially ordered set)"
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− | The [[Complete lattice|complete lattice]] | + | 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. | |
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====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> | + | <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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====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> | + | <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> | ||
+ | |||
+ | {{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) |
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