way below

Let P be a poset and a,bP. a is said to be way below b, written ab, if for any directed setMathworldPlanetmath DP such that D exists and that bD, then there is a dD such that ad.

First note that if ab, then ab since we can set D={b}, and if P is finite, we have the converseMathworldPlanetmath (since DD). So, given any element bP, what exactly are the elements that are way below b? Below are some examples that will throw some light:


  1. 1.

    Let P be the poset given by the Hasse diagram below:


    where the dotted lines denote infinite chains between the end points. First, note that every element in P is below () b. However, only v is way below b. u, for example, is not way below b, because D={xpx<b} is a directed set such that D=b and non of the elements in D are above u. This illustrates the fact that if P has a bottom, it is way below everything else.

  2. 2.

    Suppose P is a latticeMathworldPlanetmath. Then ab iff for any set D such that D exists and bD, there is a finite subset FD such that aF.


    (). Suppose ab. Let D be the set in the assumptionPlanetmathPlanetmath. Let E be the set of all finite joins of elements of D. Then DE. Also, every element of E is bounded above by D. If t is an upper bound of elements of E, then it is certainly an upper bound of elements of D, and hence Dt. So D is the least upper bound of elements of E, or E=D. Furthermore, E is directed. So there is an element eE such that ae. But e=F for some finite subset of D, and this completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath one side of the proof.

    (). Let D be a directed set such that D exists and bD. There is a finite subset F of D such that aF. Since D is directed, there is an element dD such that d is the upper bound of elements of F. So ad, completing the other side of the proof. ∎

  3. 3.

    With the above assertion, we see that, for example, in the lattice of subgroups L(G) of a group G, HK iff H is finitely generatedMathworldPlanetmathPlanetmath. Other similar examples can be found in the lattice of two-sided idealsMathworldPlanetmath of a ring, and the lattice of subspacesMathworldPlanetmath (projective geometry) of a vector space.

  4. 4.

    In particular, if P is a chain, then ab implies that ab. If D is a set such that D exists and bD, then there is a dD such that bd (otherwise b is an upper bound of elements of D and Db), so ad.

  5. 5.

    Here’s an example where ab in P but a is not the bottom of P. Take two complete infinite chains C1 and C2 with bottom 0 and 1, and let P be their productPlanetmathPlanetmathPlanetmath (http://planetmath.org/ProductOfPosets) P=C1×C2. What elements are way below (1,1)? First, take D={(a,1)0a<1}. Since P is complete, D=(1,1), but every element of D is stricly less than (1,1), so (1,1) is not way below itself. What about elements of the form (a,1), a1? If we take D={(1,b)0b<1}, then D=(1,1) once again. But no elements of D are above (a,1). So (a,1) can not be way below (1,1). Similarly, neither can (1,b) be way below (1,1). Finally, what about (a,b) for a<1 and b<1? If D is a set with D=(1,1), then D1=1 and D2=1, where D1={x(x,1)D} and D2={y(1,y)D}. Since C1 and C2 are chains, a1 implies that there is an sD1 such that as. Similarly, there is a tD2 such that bt. Together, (a,b)(s,t)D. So (a,b)(1,1).

  6. 6.

    Let X be a topological spaceMathworldPlanetmath and L(X) be the lattice of open sets in X. Suppose U,VL(X) and UV. If there is a compact subset C such that UCV, then UV.


  • In a lattice L, aa iff a is a compact element. This follows directly from the assertion above. In fact, a compact element can be defined in a general poset as an element that is way below itself.

  • If we remove the condition that D be directed in the definition above, then a is said to be way way below b.


  • 1 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, ContinuousPlanetmathPlanetmath Lattices and Domains, Cambridge University Press, Cambridge (2003).
Title way below
Canonical name WayBelow
Date of creation 2013-03-22 16:38:28
Last modified on 2013-03-22 16:38:28
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 06B35
Classification msc 06A99
Synonym way way below
Synonym way-below
Synonym way-way-below
Defines way below relationMathworldPlanetmathPlanetmath
Defines way way below relation