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Homedisjunction property of Wallman

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

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

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

The proof of this equivalence can be found in this online article.

## Mathematics Subject Classification

06A06*no label found*

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## Comments

## Alternative characterizations of Wallman's disjunction prope...

In my draft article "Filters on Posets" (http://www.mathematics21.org/binaries/filters.pdf) I found three alternative characterizations of so called "separable lattices" which are equivalent to Wallman's disjunction property for the case of meet-semilattices. This should be somehow addressed in the encyclopedia article. I deem that these results in the mentioned article are to trivial to be called research and can be added to PlanetMath. What is the opinion of the community?

--

Victor Porton - http://www.mathematics21.org

* Algebraic General Topology and Math Synthesis

* Category Theory - new concepts

## Re: Alternative characterizations of Wallman's disjunction p...

I agree.