ideal completion of a poset
Let be a poset. Consider the set of all order ideals of .
is an algebraic dcpo, such that can be embedded in.
We shall list, and when necessary, prove the following series of facts which ultimately prove the main assertion. For convenience, write .
is a poset with defined by set theoretic inclusion.
For any , .
is a dcpo. Suppose is a directed set in . Let . For any , and for some ideals . As is directed, there is such that and . So and hence there is such that and . This shows that is directed. Next, suppose and . Then for some , so as well. This shows that is a down set. So is an ideal of : .
For every , is a compact element of . If , where is directed in , then , or , which implies for some ideal . Therefore , and is way below itself: is compact.
is an algebraic dcpo. Let . Let . For any , there is such that and . This shows that and in , so that is directed. It is easy to see that . Since is a join of a directed set consisting of compact elements, is algebraic.
This completes the proof. ∎
Definition. is called the ideal completion of .
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 . Clearly, is a lower set. For every , we have such that and . Since and are both finite joins of elements of those ideals in , so is . Since and , is directed. If is any ideal larger than any of the ideals in , clearly , since is directed. So . Therefore, .
If , then , the bottom of , is the join of the empty family of ideals in . By this entry (http://planetmath.org/CriteriaForAPosetToBeACompleteLattice), is a complete lattice. ∎
If is a lower semilattice, then so is .
Let be two ideals in and . By definition, and are non-empty, so let and . As is a lower semilattice, exists and and . So , and that is non-empty. If , then or . Similarly . Therefore and is a lower set. If , then there is and such that . So and is directed. This means that . ∎
- 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).
|Title||ideal completion of a poset|
|Date of creation||2013-03-22 17:03:01|
|Last modified on||2013-03-22 17:03:01|
|Last modified by||CWoo (3771)|