All Questions
Tagged with set-theory posets
74 questions
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 ...
- 42.8k
4
votes
0
answers
153
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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
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 ...
- 41.9k
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 ...
- 32.1k
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 $\...
- 236k
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 $n\geq N$.
...
- 236k
31
votes
3
answers
2k
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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
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 ...
- 32.1k
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 ...
CommunityBot
- 1
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\...
- 327
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 ...
CommunityBot
- 1
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 ...
- 1,101
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 ...
CommunityBot
- 1
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 $P$, ...
- 7,105
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 ...
CommunityBot
- 1
3
votes
1
answer
343
views
A compactness property of posets
Consider a poset $P$ and suppose that every finite subset admits a supremum. Call an ideal $I$ of $P$ minimal infinite if it is infinite and every ideal properly contained in $I$ is finite. I am ...
- 14k
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 ...
- 32.5k
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 ...
- 98
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 ...
- 44.4k
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
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
2
votes
1
answer
362
views
Selecting k sub-posets
I ran into the following algorithmic problem while experimenting with classification algorithms. Elements are classified into a polyhierarchy, what I understand to be a poset with a single root ("...
- 2,350
2
votes
0
answers
131
views
non-degenarete tools to calculate a derived functor on a model category which is a poset?
Are there theorems (esp. computational tools) on model categories which survive and do not trivialise when its underlying category is a (quasi-)poset ? Are there tools
that may help to calculate a ...
- 1,299