All Questions
3 questions
9
votes
1
answer
501
views
Does this property of a partially ordered set have a name?
What do you call a poset with this property? For any elements $a,b,c,d$ such that $\{a,b\}\le\{c,d\}$, there is an element x such that $\{a,b\}\le x\le\{c,d\}$. (Equivalently, for any finite sets $A\...
7
votes
1
answer
357
views
Forcing axiom for a single poset
Let $FA_\kappa (\mathbb{P})$ be the claim that for every family $\mathscr{D}$ of dense sets in the poset $\mathbb{P}$ with $\vert \mathscr{D} \vert = \kappa $ there is a filter $G$ such that for all $...
4
votes
1
answer
240
views
Decomposing a poset into directed subposets
Let us say that a poset $P$ is $\mathbf{\kappa}$-directed iff every collection of fewer than $\kappa$-many elements in $P$ has an upper bound in $P$. $P$ has the $\mathbf{\kappa}$ chain condition iff ...