All Questions
Tagged with set-theory posets
10 questions
31
votes
3
answers
2k
views
Is the fixed point property for posets preserved by products?
Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point.
Theorem. Suppose $P$ and $Q$ are posets ...
- 1,872
6
votes
1
answer
223
views
Minimal Hausdorff topologies compatible with a bunch of functions
Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, \tau)\...
- 48.9k
10
votes
2
answers
2k
views
Terminology about trees
In set theory, a tree is usually defined as a partial order such that the set of elements below any given one is well-ordered. I am interested in the class of partial orders $P$ such that for every $...
- 18.6k
7
votes
1
answer
195
views
Bounding and domination numbers for relation $\leq$ modulo $\omega$-nullsets
We say that $A\subseteq \omega$ is a nullset if $$\lim\sup_{n\to \infty} \frac{|A\cap n|}{n+1} = 0.$$
Let $\omega^\omega$ denote the set of functions $f:\omega\to\omega$. We define a pre-ordering ...
- 48.9k
6
votes
2
answers
702
views
A sequence of generic filters that does not come from an iteration
Fix a countable transitive model $M$ of ZFC.
In my answer to this question I indicated that there are forcing iterations
$((Q_\alpha:\alpha\leq\omega),(\dot P_\alpha:\alpha<\omega))$ in $M$ and ...
- 16.2k
6
votes
1
answer
229
views
Smallest ordinal $\mu$ not embeddable in ${\cal P}(\omega)$
The motivation for this question is the startling fact that there is an order-preserving injective map (embedding) from $\mathbb{R}$ into ${\cal P}(\omega)$. (Think Dedekind cuts.) I am wondering how &...
- 48.9k
5
votes
0
answers
134
views
Chains of length $2^\kappa$ in ${\cal P}(\kappa)$ [duplicate]
It is a fact that continues to boggle my mind: There is a set ${\cal C}\subseteq {\cal P}(\omega)$ such that $|{\cal C}|=\frak{c}=2^{\aleph_0}$ and for all $A,B\in{\cal C}$ we have $A\subseteq B$ or $...
- 48.9k
4
votes
0
answers
435
views
Can infinite bounded distibutive lattices be "arbitrarily wide"?
I was always thinking, in an informal way, that the powerset lattices ${\cal P}(X)$ (where $X$ is an infinite set) are the "widest" bounded distributive lattices with respect to their height. (In ${\...
- 48.9k
4
votes
1
answer
227
views
Cardinality of maximal chains in the poset of ultrafilters with Rudin-Keisler ordering
Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:...
- 48.9k
3
votes
3
answers
3k
views
Well-ordered cofinal subsets [closed]
Let $(P, \leq)$ be a total ordering (some of you prefer the name linear order). Can we find a subset $R\subseteq P$ which is well ordered (with respect to $\leq\upharpoonright R$) and cofinal in $P$, ...
- 113