All Questions
44 questions
3
votes
1
answer
132
views
Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?
If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
- 48.9k
2
votes
1
answer
285
views
Size of antichains in powerset of $\mathbb N$
Take a countably infinite set $S$, say $\mathbb N$. Is it possible for there to be an antichain in $\mathcal P(S)$ (with the inclusion ordering) of continuum cardinality?
- 33
1
vote
0
answers
78
views
Set-theoretic trees with ordering between siblings
In set-theoretic trees, we have the binary relation $<_T$ (which defines the "ancestor-descendant" ordering). This relation is partial, as children are incomparable in that ordering.
...
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
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
2
votes
0
answers
92
views
Can this order relation, defined in terms of all topological spaces, be defined in terms of the reals alone?
Let $K$ be the operator monoid under composition of Kuratowski's $14$ set operators generated by topological closure $k$ and complement $c.$ Kuratowski's 1922 paper gives the poset diagram of the ...
- 870
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
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
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
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
2
votes
0
answers
114
views
Product-decomposition of ${\cal P}(\omega)/\rm{fin}$ [closed]
For $A,B\in {\cal P}(\omega)$ let us say that $A\simeq_{\rm{fin}} B$ if both $A\setminus B$ and $B\setminus A$ are finite. It is easy to see that this establishes an equivalence relation on ${\cal P}(\...
- 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
2
votes
2
answers
248
views
Maximal uncountable chains in ${\cal P}(\omega)$
Let ${\cal P}(\omega)$ denote the power-set of $\omega$. We order it by set inclusion $\subseteq$ and say that ${\cal C}\subseteq {\cal P}(\omega)$ is a chain if for all $A, B\in {\cal C}$ we have $A\...
- 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
3
votes
1
answer
193
views
${\frak b}$ and ${\frak d}$ in the Rudin-Keisler preordering
If $(Q,\leq)$ is any preordered set (that is, $\leq$ is a reflexive and transitive, but not necessarily anti-symmetric relation), then we say that $S\subseteq Q$ is
unbounded if for all $q\in Q$ ...
- 48.9k
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
1
vote
1
answer
140
views
Thinning directed sets ${\frak P}$ of partitions of $\omega$ with no ${\frak P}$-discrete subsets
This question branches from Taras Banakh's recent question on a cardinal characteristic connected to families of partitions that are directed in the ordering of partition refinement.
A partition $\...
- 48.9k
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
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
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
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
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
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
0
votes
1
answer
132
views
Upward generators of $[\omega]^\omega$
If $(P,\leq)$ is a poset and $S\subseteq P$ we let $$\uparrow S = \{p\in P: p\geq s\text{ for some }s\in S\}.$$
Let $([\omega]^\omega,\subseteq)$ denote the collection of infinite subsets of $\omega$,...
- 48.9k
0
votes
1
answer
192
views
$\text{Max}\big(\text{Sub}(\text{Sym}(\omega))\setminus \{\text{Sym}(\omega)\}\big)$
If $G$ is any group, then by $\text{Sub}(G)$ we denote the collection of all subgroups, ordered by $\subseteq$. If $(P,\leq)$ is a partially ordered set we let $\text{Max}(P)$ and the set of maximal ...
- 48.9k
3
votes
1
answer
143
views
The Wallman and interval topologies on non-principal ultrafilters with the Rudin-Keisler preorder
If $(P,\leq)$ is a pre-odered set (that is, $\leq$ is a reflexive and transitive relation) and $x\in P$, we set $(\uparrow_{\leq} x) = \{p\in P: p\geq x\}$ and $(\downarrow_{\leq} x) = \{p\in P: p\leq ...
- 48.9k
3
votes
3
answers
341
views
Countably infinite posets isomorphic to its intervals
Let $(P,\leq)$ be a countably infinite poset with the property that whenever $a<b\in P$ then $P\cong [a,b]$.
Question. If $P$ does contain elements $a,b$ with $a<b$, does this imply that $P \...
- 48.9k
4
votes
1
answer
178
views
"Gaps" in the Rudin-Keisler ordering
If $(P,\leq)$ is a poset and $p\in P$, then we say that $p$ is the lower part of a gap there is $q \in P$, $q>p$ such that $[p,q] = \{p,q\}$. (This is equivalent to the statement that $(\uparrow p) ...
- 48.9k
5
votes
1
answer
305
views
A different ordering on ${\cal P}(\omega)$
For $A, B \subseteq \omega$ we set $A \leq_{\text{inj}} B$ if there is an injective and order-preserving map $f:\omega\to \omega$ , such that $f(A)$ is a down-set of $B$. It is easy to see that $\leq_{...
- 48.9k
4
votes
1
answer
227
views
Cardinality of maximal chains in the poset of ultrafilters with Rudin-Keisler ordering
Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:...
- 48.9k
3
votes
1
answer
134
views
Unboundedness number and domination number of a poset $(P,\leq)$
Suppose $(P,\leq)$ is a poset without maximal elements. For $X\subseteq P$ we set $X^u = \{p\in P: p \geq x \text{ for all } x\in X\}$ and call this the set of upper bounds of $X$. We say that $B\...
- 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
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
3
votes
1
answer
265
views
Antichains of maximum cardinality: posets vs lattices
The following construction gives a poset such that no antichain has maximum cardinality: For $n\in\mathbb{N}\setminus\{0\}$, let "layer" $n$ consist of an antichain of $n$ points, and as for the ...
- 48.9k
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
4
votes
0
answers
153
views
Maximality with respect to having no marriage
Let $A,B\neq \emptyset$ be disjoint and suppose $G = (A\cup B, E)$ is bipartite where for all $e\in E$ we have $e\cap A \neq \emptyset\neq e\cap B$. For $a\in A$ we set $N_G(a) = \{b\in B: (\exists e\...
- 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
3
votes
1
answer
242
views
Order dimension of $\omega^\omega/(fin)$
Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we say $f\simeq g$ if and only if $\exists N \in \omega$ such that $f(n) = g(n)$ for all $n\geq N$.
...
- 48.9k
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
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
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
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
3
votes
2
answers
689
views
A problem about posets similar to Suslin's problem
Suslin's problem is:
Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$?
The answer is that it's independent of ZFC. The related ...
- 3,982