ideal completion of a poset

Let P be a poset. Consider the set Id(P) of all order ideals of P.

Theorem 1.

Id(P) is an algebraic dcpo, such that P 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 P=Id(P).

  1. 1.

    P is a poset with defined by set theoretic inclusion.

  2. 2.

    For any xP, xP.

  3. 3.

    P can be embedded in P. The function f:PP defined by f(x)=x is order preserving and one-to-one. If xy, and ax, then ay, hence xy. If x=y, we have that xy and yx, so x=y, since is antisymmetric.

  4. 4.

    P is a dcpo. Suppose D is a directed setMathworldPlanetmath in P. Let E=D. For any x,yE, xI and yJ for some ideals I,JD. As D is directed, there is KD such that IK and JK. So x,yK and hence there is zKE such that xz and yz. This shows that E is directed. Next, suppose xE and yx. Then xI for some ID, so yIE as well. This shows that E is a down set. So E is an ideal of P: D=EP.

  5. 5.

    For every xP, x is a compact element of P. If xD, where D is directed in P, then xD, or xD, which implies xI for some ideal ID. Therefore xI, and x is way below itself: x is compact.

  6. 6.

    P is an algebraic dcpo. Let IP. Let C={xxI}. For any x,yI, there is zI such that xz and yz. This shows that xz and yz in C, so that C is directed. It is easy to see that I=C. Since I is a join of a directed set consisting of compact elements, P is algebraic.

This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof. ∎

Definition. Id(P) is called the ideal completion of P.


  • In general, the ideal completion of a poset is not a complete latticeMathworldPlanetmath. It is complete in the sense of being directed complete. This is different from another type of completion, called the MacNeille completion of P, which is a complete lattice.

  • If P is an upper semilatticePlanetmathPlanetmath, then so is Id(P). In fact, the join of any non-empty family of ideals exists. Furthermore, if P has a bottom element 0, then Id(P) is a complete lattice.


    Let S be a non-empty family of ideals in P. Let A be the set of P consisting of all finite joins of elements of those ideals in S, and B=A. Clearly, B is a lower set. For every a,bB, we have c,dA such that ac and bd. Since c and d are both finite joins of elements of those ideals in S, so is cd. Since acd and bcd, B is directed. If I is any ideal larger than any of the ideals in S, clearly AI, since I is directed. So B=AI=I. Therefore, B=S.

    If 0P, then 0, the bottom of Id(P), is the join of the empty family of ideals in P. By this entry (, Id(P) is a complete lattice. ∎

  • If P is a lower semilatticePlanetmathPlanetmath, then so is Id(P).


    Let I,J be two ideals in P and K=IJ. By definition, I and J are non-empty, so let aI and bJ. As P is a lower semilattice, c:=ab exists and ca and cb. So cIJ, and that K=IJ is non-empty. If xyK, then xyI or xI. Similarly xJ. Therefore xIJ=K and K is a lower set. If r,sK, then there is uI and vJ such that r,su,v. So r,suv and K is directed. This means that IJId(P). ∎


  • 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
Canonical name IdealCompletionOfAPoset
Date of creation 2013-03-22 17:03:01
Last modified on 2013-03-22 17:03:01
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 06A12
Classification msc 06A06
Related topic LatticeOfIdeals
Defines ideal completion