# Tagged Questions

**6**

votes

**1**answer

386 views

### Order homomorphism functions on $\omega_1$

Let $\omega_1$ be the first uncountable ordinal,
same as the set of all countable ordinals.
Let $F$ be the set of all functions
$f$ from $\omega_1$ minus singleton $0$ into $\omega_1$ that
are (a) ...

**4**

votes

**1**answer

164 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 ...

**10**

votes

**2**answers

309 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 ...

**9**

votes

**1**answer

410 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 ...

**7**

votes

**1**answer

241 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 ...

**0**

votes

**3**answers

701 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 $P$, ...

**8**

votes

**1**answer

418 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 ...

**3**

votes

**1**answer

273 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 ...

**8**

votes

**2**answers

502 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**

votes

**1**answer

154 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 ...

**8**

votes

**1**answer

318 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 ...

**9**

votes

**1**answer

303 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 ...

**5**

votes

**1**answer

355 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 ...

**7**

votes

**3**answers

566 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**

votes

**2**answers

489 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 ...

**7**

votes

**2**answers

367 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 ...

**2**

votes

**1**answer

254 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**

votes

**0**answers

121 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 ...