All Questions
Tagged with set-theory order-theory
147 questions
5
votes
1
answer
146
views
Does there exist a section of $\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ that is "nearly Boolean"?
The following might be a somewhat esoteric question:
Does there exist an infinite cardinal $\kappa$ and a section $f$ of the quotient map $\pi:\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ (...
- 2,830
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
284
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.
...
6
votes
3
answers
236
views
Refinement-minimal intersecting covers
Motivation. Yesterday I was sitting idly in the train, contemplating the train network. I noticed that a lot of lines (not all) intersected, and some pairs of lines intersected in quite a few stations....
- 48.9k
12
votes
1
answer
525
views
Is the Tukey order well-founded
Consider the Tukey order restricted to directed orders of the form $(U,\supseteq)$, where $U$ is an ultrafilter on $\omega$. It is defined as follows:
For two ultrafilters $U,W$ on $\omega$, we say ...
- 123
7
votes
1
answer
579
views
Progress on determining which partial orders embed into the rationals
The following result is relatively well-known: (for example in Math StackExchange answer #37161)
For every countable linear order $(R,\prec)$, there is an $X\subseteq\mathbb Q$ such that $(R,\prec)$ ...
- 2,031
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
3
votes
2
answers
209
views
Posets such that the collection of principal down-sets does not have property ${\bf B}$
We say that a hypergraph $H=(V,E)$ has property ${\bf B}$ if there is $S\subseteq V$ such that for all $e\in E$ with $|e|>1$ we have $S\cap e \neq \emptyset \neq e \setminus S$.
Let $(P,\leq)$ be a ...
- 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
2
votes
1
answer
235
views
End elementary extension in infinitary logic of some $L_\alpha$ producing a $L_\beta$
Let $L_\alpha$ be some admissible level of the constructible hierarchy and $M \supseteq L_\alpha$ an extension of $L_\alpha$. I am looking for conditions under which $M \simeq L_\beta$. It is not ...
- 531
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
4
votes
1
answer
260
views
Is ${\cal P}(\omega)/\text{(fin)}$ a fractal poset?
If $(P,\leq)$ is a partially ordered set and $a,b\in P$ we set $[a,b]:=\{x\in P: a\leq x\leq b\}$. We say that $P$ is fractal if whenever $a,b\in P$ and $[a,b]$ contains more than one element, then $[...
- 48.9k
-3
votes
1
answer
101
views
Order-embeddability of ${\frak b}$ and ${\frak d}$ in $\mathbb{R}$ [duplicate]
The starting point of this question is the observation that in ${\sf (ZFC)}$, all ordinals $\alpha < \omega_1$ can be order-embedded in $\mathbb{R}$.
Let $\omega^\omega$ denote the set of all ...
- 48.9k
11
votes
1
answer
670
views
Do all toposes satisfy the internal Zorn's lemma?
I came up with this question when trying to give a more detailed answer to a question by Tim Campion in a comment to Ingo Blechschmidt's answer to Examples of statements that are valid in every ...
- 18.4k
24
votes
1
answer
2k
views
Why do we need "canonical" well orders?
(I asked this question on Math.SE earlier but received no response and am therefore moving it here, please note that I realise this question is probably incredibly naïve for the experienced set-...
- 343
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
4
votes
1
answer
209
views
Exactly how much (and how little) can partial ordered sets (classes) embed to the cardinalities
In the paper "Convex Sets of Cardinals", Truss mentioned a result of Jech:
If $M$ is a countable transitive model of ZFC, and $(P,<)∈M$ is a poset, then there exists a Cohen extension of ...
- 1,676
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
10
votes
0
answers
381
views
Can one define in ZFC a directed system of embeddings on the class of all linear orders realizing the surreal line as the direct limit?
Consider the surreal line $\langle\newcommand\No{\text{No}}\No,\leq\rangle$, in its order structure only. This is a proper class linear order, which is universal for all set-sized linear orders, as ...
- 236k
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
7
votes
2
answers
496
views
Counterexample for Chvatal's conjecture in an infinite set
Let $X \neq \emptyset$ be a set. We say that ${\cal F} \subseteq {\cal P}(X)$ is a down-set if ${\cal F}$ is closed under taking subsets. Whenever $a \in X$, we let ${\cal F}_a = \{ S \in F : a \in S\}...
- 48.9k
5
votes
1
answer
270
views
Question about a family of nested countable subsets of $\mathbb{R}$
Let $\mathcal{F}$ denote a family of countable subsets of $\mathbb{R}$, such that for each $U, V\in\mathcal{F}$ we have that $U\subseteq V$, or $V\subseteq U$. Let $(\mathcal{F}, \preceq)$ denote the ...
- 291
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
197
views
Weak compactness is to trees as [?] is to lattices?
Let $\kappa$ be an inaccessible cardinal. Recall that $\kappa$ is weakly compact if every tree of height $\kappa$ has either a level of size $\kappa$ or a branch of size $\kappa$.
So if $\kappa$ is a ...
- 63.9k
2
votes
1
answer
136
views
Is the set of "endomorphisms" of a directed set again a directed set?
Let $(I,\leq)$ be a directed set, that is $\leq$ is reflexive and transitive and for every $a,b\in I$ we find $c\in I$ such that $a,b\leq c$. Now consider the set $M$ consisting of all maps $\sigma:I\...
- 696
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
6
votes
1
answer
256
views
Maximal independent sets in MAD families
We call ${\cal A}\subseteq {\cal P}(\omega)$ almost disjoint if ${\cal A}\neq \varnothing$, every member of ${\cal A}$ is infinite, and for $A_1\neq A_2\in {\cal A}$ we have that $A_1\cap A_2$ is ...
- 48.9k
4
votes
0
answers
135
views
Cofinality without choice: can this coarse definition suffer badly?
This is a rephrased version of a question previously asked at MSE without success.
Working in $\mathsf{ZF}$, it is no longer possible in general to give every linear order an ordinal cofinality. For ...
- 21.2k
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
1
vote
1
answer
143
views
Is the Rudin-Keisler ordering a continuous relation?
If $X, Y$ are topological, and $R\subseteq X\times Y$ we say that $R$ is continuous (from $X$ to $Y$) if for every $V\subseteq Y$ with $V$ open, we have $$R^{-1}(V) = \{u\in U: \exists v\in V:(u,v)\...
- 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 ...
- 12.5k
5
votes
1
answer
321
views
Complete Boolean algebras of subsets of $\mathbb N$
Let $\mathfrak A$ be a subset of $\mathrm{Pow}(\mathbb N)$, the powerset of $\mathbb N$. Assume that $\mathfrak A$ is a complete Boolean algebra in the induced order, i.e., the inclusion order. Does ...
- 974
11
votes
0
answers
256
views
Existence of a strong antichain
Call an antichain (set of pairwise incomparable elements) $A$ of a poset $P$ strong if for every $p,q \in P$ with $p \leq q$ there exists an $a\in A$ which is comparable with both $p$ and $q$.
...
- 111
6
votes
1
answer
242
views
A monotone countably unbounded function from $\omega^\omega$ to $\omega^{\omega_1}$
For a set $X$ we endow the set $\omega^X$ of all functions from $X$ to $\omega$ with the natural partial order $\le$ defined by $f\le g$ iff $f(x)\le g(x)$ for all $x\in X$. A function $\mu:\omega^\...
- 41.8k
11
votes
1
answer
418
views
A monotone countably cofinal function from $\omega^\omega$ to $\omega^{\omega_1}$
For a set $X$ we endow the set $\omega^X$ of all functions from $X$ to $\omega$ with the natural partial order $\le$ defined by $f\le g$ iff $f(x)\le g(x)$ for all $x\in X$.
A function $f:\omega^\...
- 41.8k
5
votes
1
answer
174
views
Preserve unbounded sets between different cofinality
Working in ZFC, let $\kappa,λ$ be cardinals with $\kappa>λ$, and assume that $\kappa$ is regular.
We say that a function $F:\kappa^n→λ$, for some finite $n$, is preserving unbound, if for all $a⊆\...
- 1,676
17
votes
1
answer
1k
views
Has the exponentiation of ordinals a nice geometric model?
It is well known that the sum $\alpha+\beta$ of two ordinals $\alpha,\beta$ can be defined "geometrically" as the order type of the sum $(\{0\}\times \alpha)\cup(\{1\}\times\beta)$ endowed ...
- 41.8k
5
votes
1
answer
356
views
Calculate the $\downarrow$, $\downarrow\uparrow$ and $\uparrow\downarrow$ cofinalities of the poset of nontrivial finitary partitions of $\omega$
Let $(P,\le)$ be a poset. For a point $x\in P$ let
$${\downarrow}x=\{p\in P:p\le x\}\quad\text{and}\quad{\uparrow}x=\{p\in P:x\le p\}$$be the lower and upper sets of the point $x$, and for a subset $...
- 41.8k
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
2
votes
3
answers
537
views
Consistency of embedding cardinals in linear orderings
Background
The fact that there is no suborder of $\mathbb R$ which is of type $\omega_1$ suggests (to me) that the continuum $c$ cannot be very far from $\omega_1$: How could $c$ be far away from $\...
- 2,467
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
2
votes
2
answers
467
views
(Types of) induction on infinite chains
This question may be trivial, or overly optimistic. I do not know (but I guess the latter...). I am a group theorist by trade, and the set-up I describe cropped up in something I want to prove. So ...
4
votes
1
answer
774
views
Is this lemma equivalent to the axiom of choice?
Given any pre-ordering $\preceq$ of an arbitrary set $X$ is the following lemma:
$$\text{There exists an inclusion minimal set }S\text{ satisfying }\{a\preceq b:b\in S\}=X\\\iff \text{ Every chain in ...
- 2,506
1
vote
0
answers
97
views
Generalization of the linear extension theorem to directed acyclic graphs
Using Zorn's lemma one can prove a generalization of the order extension theorem, that states any acyclic digraph is always contained in another acyclic unilaterally connected digraph on the same ...
- 2,506