disjunction property of Wallman


A partially ordered setMathworldPlanetmath š”„ with a least element 0 has the disjunction property of Wallman if for every pair (a,b) of elements of the poset, either bā‰¤a or there exists an element cā‰¤b such that cā‰ 0 and c has no nontrivial common predecessor with a. That is, in the latter case, the only x with xā‰¤a and xā‰¤c is x=0.

For the case if the poset š”„ is a āˆ©-semilattice disjunction property of Wallman is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to every of the following three formulasMathworldPlanetmathPlanetmath:

  1. 1.

    āˆ€a,bāˆˆš”„:({cāˆˆš”„|cāˆ©aā‰ 0}={cāˆˆš”„|cāˆ©bā‰ 0}ā‡’a=b);

  2. 2.

    āˆ€a,bāˆˆš”„:({cāˆˆš”„|cāˆ©aā‰ 0}āŠ†{cāˆˆš”„|cāˆ©bā‰ 0}ā‡’aāŠ†b);

  3. 3.

    āˆ€a,bāˆˆš”„:(aāŠ‚bā‡’{cāˆˆš”„|cāˆ©aā‰ 0}āŠ‚{cāˆˆš”„|cāˆ©bā‰ 0}).

The proof of this equivalence can be found in http://www.mathematics21.org/binaries/filters.pdfthis online article.

Title disjunction property of Wallman
Canonical name DisjunctionPropertyOfWallman
Date of creation 2013-03-22 17:53:48
Last modified on 2013-03-22 17:53:48
Owner porton (9363)
Last modified by porton (9363)
Numerical id 7
Author porton (9363)
Entry type Definition
Classification msc 06A06
Synonym Wallmanā€™s disjunction property
Related topic Poset