disjunction property of Wallman

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

For the case if the poset 𝔄 is a ∩-semilattice disjunction property of Wallman is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to every of the following three formulasMathworldPlanetmathPlanetmath:

  1. 1.

    βˆ€a,bβˆˆπ”„:({cβˆˆπ”„|c∩aβ‰ 0}={cβˆˆπ”„|c∩bβ‰ 0}β‡’a=b);

  2. 2.

    βˆ€a,bβˆˆπ”„:({cβˆˆπ”„|c∩aβ‰ 0}βŠ†{cβˆˆπ”„|c∩bβ‰ 0}β‡’aβŠ†b);

  3. 3.

    βˆ€a,bβˆˆπ”„:(aβŠ‚bβ‡’{cβˆˆπ”„|c∩aβ‰ 0}βŠ‚{cβˆˆπ”„|c∩bβ‰ 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 Wallman’s disjunction property
Related topic Poset