All Questions
Tagged with set-theory posets
74 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
28
votes
1
answer
6k
views
What is the cofinality of the co-infinite subsets of ${\bf N}$?
Let ${\mathcal A}$ be the family of subsets $A$ of the natural numbers ${\mathbf N}$ which are co-infinite (i.e., their complement is infinite). We partially order this family by set inclusion. A ...
- 114k
19
votes
1
answer
1k
views
Suprema of directed sets
Let $(X, \le)$ be a partially ordered set. We call a subset $S \subseteq X$...
... a chain if each two elements in $S$ are comparable with respect to $\le$ (in other words, $S$ is linearly ordered ...
- 12.6k
17
votes
1
answer
1k
views
How is this fixed point theorem related to the axiom of choice?
I'm hoping the answer to this is well-known.
Let $X$ be an ordered set (i.e. poset). An inflationary operator $f$ on $X$ is a function $f: X \to X$, not necessarily order-preserving, such that $f(x) \...
- 27.7k
16
votes
1
answer
627
views
Universal locally countable partial order
Call a poset locally countable if the set of predecessors of every member of the poset is countable. Is the following consistent?
There is no locally countable poset $P$ of size continuum such that ...
- 9,641
14
votes
3
answers
1k
views
Order homomorphism functions on $\omega_1$
I posted the following question more than two years ago on MO (and then reposted on MSE), but the answer remains incomplete, so I thought I would rephrase it a bit (to make the statement clearer) and ...
- 1,375
13
votes
1
answer
283
views
Can any poset of cardinality $\leq 2^{\aleph_0}$ be embedded in ${\cal P}(\omega)/(\text{fin})$?
We endow ${\cal P}(\omega)$ with an equivalence relation by saying that $A\simeq_{\text{fin}} B$ iff the symmetric difference $A\Delta B$ is finite. The resulting set of equivalence classes is denoted ...
- 48.9k
12
votes
1
answer
642
views
Posets preserving stationary subsets of $\omega_1$ and no new $\text{cof}(\omega)$ ordinals, but without countable covering property
What are some examples of posets $\mathbb{P}$ which have the following properties? It's OK if the definition uses large cardinals or some other hypothesis.
$\mathbb{P}$ preserves stationary subsets ...
- 2,231
10
votes
4
answers
383
views
Universal poset for cardinals $\kappa \geq \aleph_0$
Given a cardinal $\kappa\geq \aleph_0$, is there a poset $(P,\leq)$ with $|P| = \kappa$ such that every poset of cardinality $\kappa$ can be order-embedded into $(P,\leq)$?
- 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
10
votes
2
answers
469
views
Do operations generate well-ordered sets only?
I've read @TauMu's question about the set of functions $\mathbb N\rightarrow\mathbb N$ generated from the identity map by repeatedly applying exponentiation of two already ...
- 7,500
10
votes
1
answer
707
views
Does "antichain" mean something different in set-forcing than in lattice theory?
On page 3 of Introduction to Lattices and Order, Davey and Priestley define an antichain in a poset $\langle P,\leq\rangle$ as a set of pairwise incomparable elements:
The ordered set P is an ...
- 3,267
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\...
user33772
9
votes
1
answer
463
views
Set of maximal subfields not containing particular elements.
Instead of extending a field, by adjoining a new element, consider what happens if we remove an element or elements.
This started as a question on math.SE Field reductions where Pete L. Clark ...
- 301
9
votes
1
answer
984
views
Ordered sum of posets
Let $I$ be a poset and for any $i$ let $P_i$ be a poset. Let $P$ be the sum over $I$ of the sets $P_i$, and let $<_P$ be the relation defined on $P$ by $q<_Pr$ iff $q$ and $r$ are members of the ...
- 2,655
9
votes
0
answers
250
views
Distributivity of certain infinite products
Suppose we have a sequence of posets $\{\mathbb P_n : n\in\omega\}$ such that for each $n$, $\mathbb P_{n+1}$ is $|\mathbb P_n|^+$-distributive. Is $\prod_{n>0} \mathbb P_n$ necessarily $|\mathbb ...
- 18.6k
8
votes
1
answer
275
views
Is the measurable space $(\omega_1,\mathcal{P}(\omega_1))$ separable?
Here $\omega_1$ is the first uncountable ordinal, and $\mathcal{P}(\omega_1)$ denotes the power set of $\omega_1$. Separable means countably generated as a $\sigma$-algebra.
- 515
8
votes
1
answer
232
views
Introducing meets while preserving directed closure
A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound.
Question: Suppose $\mathbb{P}$ is a separative partial order which is $\...
- 2,231
8
votes
1
answer
415
views
Decomposing posets into countably many chains
A conjecture of Galvin's is that the following is possible (which I take to mean that the consistency of the following can be proven relative to the consistency of something like ZFC, or ZFC plus some ...
- 3,982
7
votes
2
answers
344
views
Surjective order-preserving map $f:{\cal P}(X)\to \text{Part}(X)$
Let $X$ be a set, and let $\text{Part}(X)$ denote the collection of all partitions of $X$. For $A, B\in \text{Part}(X)$ we set $A\leq B$ if $A$ refines $B$, that is for all $a\in A$ there is $b\in B$ ...
- 48.9k
7
votes
3
answers
1k
views
Characterizing forcings that don't add any dominating reals
Regarding reals as functions from $\omega$ to $\omega$, let's say a real $f$ eventually dominates $g$ iff $(\exists n)(\forall m > n)[ f(m) > g(m)]$. Let's say that a (non-trivial separative) ...
- 3,982
7
votes
2
answers
845
views
Statements forced by one condition of a poset, but not the whole thing
In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be ...
- 3,982
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 $...
- 505
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
7
votes
1
answer
368
views
Which of these relations on partial orders allows us to identify forcing equivalence?
Background
This question was inspired by Justin Palumbo's excellent question Cantor Bernstein for notions of forcing.
In his question, Justin considers a relation $\lhd$ on partial orders (defined ...
- 1,146
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
2
answers
1k
views
Poset dimension and width (Dilworth's theorem)
For a given poset $P$, let $\mathrm{dim}(P)$ denote the least cardinal $\kappa$ such that there exists a $\kappa$-sized collection of linear extensions of $P$, say $\mathcal{L}$, such that $\leq_P = \...
- 1,006
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
6
votes
1
answer
356
views
Is every homogeneous poset a lattice?
A poset $(P,\leq)$ is homogeneous if $P\cong [a,b]$ for all $a,b\in P$ with $a<b$ (where $[a,b] := \{x\in P: a\leq x\leq b\}$).
Examples of homogeneous posets include $[0,1]$, $[0,1]\cap \mathbb{Q}$...
- 48.9k
6
votes
1
answer
256
views
Poset as union of posets of lower cofinality
Let $ \mathbb{P}$ be any directed, well-founded poset of cofinality $ \aleph_{n+1}$, where $n$ is a natural.
Can we write it as an increasing union $ \mathbb{P} = \bigcup_{\alpha < \omega_{n+1} } \...
- 505
6
votes
1
answer
153
views
Topologies with no minimal $T_2$ topologies above them
Let $(X,\tau)$ be a topological space. With $T_2(\tau)$ we denote the collection of $T_2$-topologies on $X$ that contain $\tau$.
Is there an example of a topology $\tau$ such that the partially ...
- 48.9k
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
6
votes
1
answer
213
views
Pairwise non-isomorphic interval-isomorphic lattices
Let us call a lattice $(L,\leq)$ interval-isomorphic if for all $a<b \in L$ we have $L \cong [a,b]$, where $[a,b]=\{x\in L:a\leq x\leq b\}$.
Are there $2^{\aleph_0}$ pairwise non-isomorphic ...
- 48.9k
6
votes
0
answers
170
views
Katetov ordering on ideals on $\omega$
Recall that a nonempty set ${\cal I}\subseteq {\cal P}(\omega)$ is a (set) ideal if
$B\in{\cal I}$ and $A\subseteq B$ imply $A\in{\cal I}$, and
$A,B \in {\cal I}$ implies $A\cup B\in {\cal I}$.
By $\...
- 48.9k
5
votes
1
answer
298
views
Partition induced by a cover
Let $X$ be a set and let $(Y_i)_{i \in I}$ be a family of (not necessarily pairwise disjoint) subsets covering $X$,
$$ X = \bigcup_{i\in I} Y_i.$$
For any subset $J \subseteq I$, we then define
$$ Y_J ...
- 9,853
5
votes
1
answer
215
views
Cofinal well-founded subset in mod finite order
The mod finite order on ${}^\omega \omega$ is defined as $f \leq^\ast g$ if and only if $f(n) \leq g(n)$ except for finitely many $n \in \omega$. My question is: can we always extract a cofinal well-...
- 505
5
votes
1
answer
305
views
A different ordering on ${\cal P}(\omega)$
For $A, B \subseteq \omega$ we set $A \leq_{\text{inj}} B$ if there is an injective and order-preserving map $f:\omega\to \omega$ , such that $f(A)$ is a down-set of $B$. It is easy to see that $\leq_{...
- 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
2
answers
291
views
The average size of downward closed family of the subsets of $[n]$ is at most $n/2$?
I learned that the average size in any ideal of subsets of $[n]$ is at most $n/2$, but I think the downward closed family of the subsets of $[n]$ also satisfied. I want to know how to proof it or it ...
- 159
4
votes
2
answers
191
views
Ordinal-universal linear order on $\kappa$ elements
The starting point of this question is the observation that if $\lambda$ is a countable ordinal, then there is an order-embedding $e:\lambda \hookrightarrow \mathbb{Q}$.
Given an infinite cardinal $\...
- 48.9k
4
votes
1
answer
239
views
Cofinal rectangles in poset
Suppose $(P, <)$ is a poset of cofinality $\aleph_2$ and additivity (least cardinality of an unbounded subset) $\aleph_1$. Can we conclude the existence of a cofinal subset of order-type $\omega_1 \...
- 505
4
votes
1
answer
221
views
Embedding ordinals with the order topology into connected $T_2$-spaces
Is there a limit ordinal $\kappa_0$ with $\kappa_0 \lt 2^{\aleph_0}$ and such that for every limit ordinal $\lambda$ with $\kappa_0\leq \lambda\lt 2^{\aleph_0}$ there is a connected $T_2$-space $X_\...
- 48.9k
4
votes
1
answer
178
views
"Gaps" in the Rudin-Keisler ordering
If $(P,\leq)$ is a poset and $p\in P$, then we say that $p$ is the lower part of a gap there is $q \in P$, $q>p$ such that $[p,q] = \{p,q\}$. (This is equivalent to the statement that $(\uparrow p) ...
- 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
4
votes
1
answer
120
views
Searching for cofinal subsets of directed sets subject to finite constraints
Let $(P,\leq)$ be a directed set with uncountable cofinality. For every element $p\in P$, we are given a finite set $c_p\subset P\smallsetminus \{p\}$ of "incompatible elements". We say that ...
- 380
4
votes
1
answer
289
views
Does the lattice of partitions map onto the lattice of subsets?
Let $X\neq \emptyset$ be a set and let $X^X$ denote the collection of all functions $f:X\to X$. We put a binary relation (reflexive and transitive), the composition preorder on $X^X$ by setting for $f,...
- 48.9k
4
votes
1
answer
286
views
About subposet of Levy collapse
Let $\lambda>\kappa$ and $\operatorname{Coll}(\kappa, \lambda)$ be the poset collapsing $\lambda$ to $\kappa$. Pick a subposet $P$ which is $\lt\kappa$-closed and of size $\lambda$. Can we say that ...
user46687
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 ...
- 3,982
4
votes
0
answers
300
views
Is the set of approximating sequences for irrationals dominating?
Let $r\in [0,1]\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\mathbb{N}_+$ let $$\alpha_r(n)=\min\{|r-\frac{...
- 48.9k
4
votes
0
answers
160
views
Finite pre-orders embeddable in the Rudin-Keisler ordering
$\DeclareMathOperator{\NPU}{\operatorname{NPU}}\DeclareMathOperator{\RK}{\,\mathrm{RK}}$A pre-ordered set is a pair $(P, \leq)$ where $P$ is a set and $\leq\subseteq P\times P$ is a reflexive and ...
- 48.9k