PlanetMath: ideal completion of a poset

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ideal completion of a poset

(Definition)

Let be a poset. Consider the set $\operatorname{Id}(P)$ of all order ideals of .

Theorem 1 $\operatorname{Id}(P)$ is an algebraic dcpo, such that can be embedded in.

Proof. We shall list, and when necessary, prove the following series of facts which ultimately prove the main assertion. For convenience, write $P'=\operatorname{Id}(P)$ .

is a poset with $\le$ defined by set theoretic inclusion.
For any $x\in P$ , $\downarrow\!\!x\in P'$ .
can be embedded in . The function $f:P\to P'$ defined by $f(x)=\downarrow\!\!x$ is order preserving and one-to-one. If $x\le y$ , and $a\le x$ , then $a\le y$ , hence $\downarrow\!\!x\subseteq \downarrow\!\!y$ . If $\downarrow\!\!x=\downarrow\!\!y$ , we have that $x\le y$ and $y\le x$ , so , since $\le$ is antisymmetric.
is a dcpo. Suppose is a directed set in . Let $E=\bigcup D$ . For any $x,y\in E$ , $x\in I$ and $y\in J$ for some ideals $I,J\in D$ . As is directed, there is $K\in D$ such that $I\subseteq K$ and $J\subseteq K$ . So $x,y\in K$ and hence there is $z\in K\subseteq E$ such that $x\le z$ and $y\le z$ . This shows that is directed. Next, suppose $x\in E$ and $y\le x$ . Then $x\in I$ for some $I\in D$ , so $y\in I\subseteq E$ as well. This shows that is a down set. So is an ideal of : $\bigvee D=E\in P'$ .
For every $x\in P$ , $\downarrow\!\!x$ is a compact element of . If $\downarrow\!\!x\le \bigvee D$ , where is directed in , then $\downarrow\!\!x\subseteq \bigcup D$ , or $x\in \bigcup D$ , which implies $x\in I$ for some ideal $I\in D$ . Therefore $\downarrow\!\!x\subseteq I$ , and $\downarrow\!\!x$ is way below itself: $\downarrow\!\!x$ is compact.
is an algebraic dcpo. Let $I\in P'$ . Let $C=\lbrace \downarrow\!\!x\mid x\in I\rbrace$ . For any $x,y\in I$ , there is $z\in I$ such that $x\le z$ and $y\le z$ . This shows that $\downarrow\!\!x\le \downarrow\!\!z$ and $\downarrow\!\!y\le \downarrow\!\!z$ in , so that is directed. It is easy to see that $I=\bigvee C$ . Since is a join of a directed set consisting of compact elements, is algebraic.

This completes the proof. $\qedsymbol$

Definition. $\operatorname{Id}(P)$ is called the ideal completion of .

Remarks.

In general, the ideal completion of a poset is not a complete lattice. It is complete in the sense of being directed complete. This is different from another type of completion, called the MacNeille completion of , which is a complete lattice.
If is an upper semilattice, then so is $\operatorname{Id}(P)$ . In fact, the join of any non-empty family of ideals exists. Furthermore, if has a bottom element 0, then $\operatorname{Id}(P)$ is a complete lattice.
Proof. Let be a non-empty family of ideals in . Let be the set of consisting of all finite joins of elements of those ideals in , and $B=\downarrow\!\!A$ . Clearly, is a lower set. For every $a,b\in B$ , we have $c,d\in A$ such that $a\le c$ and $b\le d$ . Since and are both finite joins of elements of those ideals in , so is $c\vee d$ . Since $a\le c\vee d$ and $b\le c\vee d$ , is directed. If is any ideal larger than any of the ideals in , clearly $A\subseteq I$ , since is directed. So $B=\downarrow\!\!A\subseteq \downarrow\!\!I=I$ . Therefore, $B=\bigvee S$ .
If $0\in P$ , then $\langle 0\rangle$ , the bottom of $\operatorname{Id}(P)$ , is the join of the empty family of ideals in . By this entry, $\operatorname{Id}(P)$ is a complete lattice. $\qedsymbol$
If is a lower semilattice, then so is $\operatorname{Id}(P)$ .
Proof. Let be two ideals in and $K=I\cap J$ . By definition, and are non-empty, so let $a\in I$ and $b\in J$ . As is a lower semilattice, $c:=a\wedge b$ exists and $c\le a$ and $c\le b$ . So $c \in I\cap J$ , and that $K=I\cap J$ is non-empty. If $x\le y\in K$ , then $x\le y\in I$ or $x\in I$ . Similarly $x\in J$ . Therefore $x\in I\cap J=K$ and is a lower set. If $r,s\in K$ , then there is $u\in I$ and $v\in J$ such that $r,s\le u,v$ . So $r,s\le u\wedge v$ and is directed. This means that $I\cap J\in \operatorname{Id}(P)$ . $\qedsymbol$

Bibliography

1: G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).

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