# disjunction property of Wallman

A partially ordered set $\mathfrak{A}$ with a least element $0$ has the disjunction property of Wallman if for every pair $(a,b)$ of elements of the poset, either $b\leq a$ or there exists an element $c\leq b$ such that $c\neq 0$ and $c$ has no nontrivial common predecessor with $a$. That is, in the latter case, the only $x$ with $x\leq a$ and $x\leq c$ is $x=0$.

For the case if the poset $\mathfrak{A}$ is a $\cap$-semilattice disjunction property of Wallman is equivalent to every of the following three formulas:

1. 1.

$\forall a,b\in\mathfrak{A}:(\{c\in\mathfrak{A}|c\cap a\neq 0\}=\{c\in\mathfrak% {A}|c\cap b\neq 0\}\Rightarrow a=b)$;

2. 2.

$\forall a,b\in\mathfrak{A}:(\{c\in\mathfrak{A}|c\cap a\neq 0\}\subseteq\{c\in% \mathfrak{A}|c\cap b\neq 0\}\Rightarrow a\subseteq b)$;

3. 3.

$\forall a,b\in\mathfrak{A}:(a\subset b\Rightarrow\{c\in\mathfrak{A}|c\cap a% \neq 0\}\subset\{c\in\mathfrak{A}|c\cap b\neq 0\})$.

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

Title disjunction property of Wallman DisjunctionPropertyOfWallman 2013-03-22 17:53:48 2013-03-22 17:53:48 porton (9363) porton (9363) 7 porton (9363) Definition msc 06A06 Wallman’s disjunction property Poset