All Questions
Tagged with set-theory boolean-algebras
80 questions
6
votes
1
answer
286
views
completions of regular suborders
Suppose $\mathbb{P}$ is a regular suborder of the separative partial order $\mathbb{Q}$ (see below for definitions). Must there always exist some complete boolean algebra $\mathbb{B}$ such that:
$\...
- 2,231
16
votes
1
answer
607
views
The dominating number $\mathfrak{d}$ and convergent sequences
All spaces considered below are compact Hausdorff.
If $K$ is a space, then $w(K)$ is its weight. For a Boolean algebra $\mathcal{A}$, $K_\mathcal{A}$ denotes its Stone space. I am interested in ...
- 1,151
4
votes
0
answers
207
views
What algebraic identities does the iteration of forcing operation satisfy?
Let $G$ be the set of all formulas $\phi(x)$ in the language of such that $ZFC\vdash\exists x\phi(x)$ exists, $ZFC\vdash\phi(x)\rightarrow``x\,\textrm{is a complete Boolean algebra}"$, $ZFC\vdash``\...
3
votes
2
answers
537
views
Proving results about complete Boolean algebras in ZFC using Boolean valued models
I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing.
I am mainly ...
7
votes
2
answers
469
views
centeredness in forcing iterations
Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$, and $B$ is $\kappa$-centered. Let $G \subset A$ be a generic ultrafilter. Is $B/G$ $\kappa$-centered in $V[G]$?
Naively, we ...
- 18.6k
8
votes
1
answer
570
views
On $V$-decisive and weakly homogeneous forcings
Suppose that $\Bbb P$ is a forcing in $V$, we say that $\Bbb P$ is $V$-decisive if whenever $\varphi(x_1,\ldots,x_n)$ is a statement in the language of forcing, and $u_1,\ldots,u_n\in V$ then $1_{\Bbb ...
- 39.7k
3
votes
1
answer
218
views
Boolean completion (of a forcing notion) isomorphic to each of its cones
Suppose $ \mathbb{P} := (P, {\leq_P}, 1_P) $ is a separative partial order. Let $ \mathbb{B} := \operatorname{RO}(\mathbb{P}) $ denote the Boolean completion.
Fix some dense embedding $ i \colon P \...
- 103
18
votes
4
answers
2k
views
Complete Boolean algebra not isomorphic to a $\sigma$-algebra
Does there exist a complete Boolean algebra that is not isomorphic to any $\sigma$-algebra? If so, what is an easy or canonical example or construction?
- 24.8k
6
votes
1
answer
934
views
The universal algebra of a $\sigma$-algebra
I am searching for the 'dual' algebraic structure of a $\sigma$-algebra. The notion of duality is like in the case of the Boolean algebra and set algebra.
If $X$ is a set, the complement and ...
- 191
10
votes
0
answers
759
views
Full conditional probabilities and versions of AC?
A probability is a finitely additive measure on a boolean algebra with total measure $1$.
A function $P:\scr B \times (\scr B - \{ 0 \})$ is a full conditional probability on $\scr B$ (for a boolean ...
13
votes
1
answer
492
views
Nontrivially nontrivial automorphisms of $P(\omega_1)/$fin
Velickovic proved (Theorem 4.1 of OCA and automorphisms of $\mathcal{P}(\omega)/\mathrm{fin}$) that, assuming
OCA (Open Coloring Axiom) and
$\rm MA_{\aleph_1}$,
every (Boolean algebra) automorphism ...
- 1,336
9
votes
1
answer
287
views
Independent families versus generators
I asked this question on M.SE a while ago and got no answers, so I'm asking it here.
Let $\kappa$ be an infinite cardinal. A family $\mathcal{A}\subseteq\mathcal{P}(\kappa)$ is independent if for ...
- 1,336
5
votes
2
answers
398
views
How complete is $\mathcal P(\kappa)/J_{bd}$?
While it is true that $\mathcal P(\kappa)$ is a complete Boolean algebra, it is not necessarily true that $\mathcal P(\kappa)/I$ is complete for an ideal $I$. In particular if we consider $I=J_{bd}$ ...
- 39.7k
10
votes
0
answers
514
views
Existence of a regular subposet which collapses everything except the top cardinal
Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular $\kappa < \delta$)...
- 2,231
7
votes
2
answers
494
views
subalgebra of a simple forcing
Let $\alpha > 0$ be any ordinal. Consider the forcing $Add(\omega,\alpha) * Add(\omega_1,1)$. Let $B$ be its boolean completion. Let $\dot{X}$ be the canonical $B$-name for the generic subset of ...
- 18.6k
7
votes
2
answers
510
views
Suslin algebras
A Suslin algebra is a generalization of a Suslin tree: it is a ccc, $(\omega,\infty)$-distributive boolean algebra. Jech proved that every Suslin algebra has size at most $2^{\omega_1}$. A proof is ...
- 18.6k
3
votes
1
answer
164
views
Algebras with countable chains only
Is there an example of an uncountable Boolean algebra $B$ in which every chain is countable and such that $\ell_\infty$ embeds into the Banach space $C(\mbox{Stone }B)$? The latter requirement is not ...
- 387
3
votes
1
answer
168
views
Independent families and chains
My question will be very short.
Suppose we have a Boolean algebra $B$ which admits an uncountable independent family. Does it follow that there is an uncountable chain of elements in $B$?
...
- 387
4
votes
2
answers
625
views
The category of Boolean-valued models associated to a model of ZFC
This is related to my previous question here, and indirectly motivated by Andreas Blass intriguing answer therein:
start from a ZF transitive model $M$, and consider the category $CBA(M)$ of ...
- 7,853
7
votes
3
answers
437
views
Chain conditions in quotients of power sets
Several days ago a friend asked me the following:
We know that in $\mathcal P(\mathbb N)$ we can find a family of size continuum that every [distinct] two intersect in a finite set. Can we do that ...
- 39.7k
5
votes
3
answers
513
views
Maximal ideals in Boolean algebras; reference request
An old theorem of Pospisil asserts that for any infinite set $I$ the power-set algebra $\wp(I)$ has $\exp \exp |I|$ many maximal ideals containing the ideal of finite sets. This result is published in ...
- 11.3k
19
votes
2
answers
4k
views
Can we put a probability measure on every $\sigma$-algebra?
The following question has puzzled me for some time:
Let $(\Omega,\Sigma)$ be a nonempty,
measurable space. Does there
necessarily exist a probability
measure $\mu:\Sigma\to[0,1]$?
If there ...
2
votes
1
answer
220
views
Extending BAs to weakly countably distributive algebras.
Suppose $\mathscr{A}$ is a complete Boolean algebra (assume it is c.c.c. if you wish). Is there any, say, canonical embedding $\mathscr{A}\subseteq \mathscr{B}$ into a complete Boolean algebra which ...
- 55
8
votes
1
answer
365
views
Counting copies of a BA within a BA: arbitrarily many vs infinitely many
Informally, I am wondering if a Boolean algebra $\mathcal{B}$ contains infinitely many disjoint copies of a Boolean algebra $\mathcal{A}$ whenever it contains arbitrarily many disjoint copies of $\...
- 596
6
votes
2
answers
1k
views
An exercise in Jech's Set Theory
I had a hard time trying to solve exercise 7.24 in Jech's book (3rd edition, 2003) and finally came to the conclusion that the result there, which should be proved might be wrong. The claim goes like ...
- 2,180
4
votes
2
answers
1k
views
density of boolean algebras
For a boolean algebra B, let d(B) be the least cardinality of a dense subset of B. Let A be a (non-regular) subalgebra of a boolean algebra B. Is it possible that d(A) > d(B)? What if d(B) = $\...
- 18.6k
5
votes
2
answers
1k
views
Free product of Boolean algebras
Given a family of Boolean algebras $\mathcal{B}=\{B_i\colon i\in I\}$ with respective Stone spaces $S_i$, the algebra of clopen (both closed and open) subsets of the product space $\textstyle\prod_{i\...
- 11.3k
12
votes
5
answers
2k
views
Jonsson Boolean algebras?
Let us say that a mathematical structure of cardinality $\omega_1$ is Jonsson whenever every one of its proper substructures is countable.
There are examples of Jonsson groups due to Shelah or ...
- 11.3k
8
votes
4
answers
2k
views
Terminology for relation on sets
Does the following relation between sets have a name or any special properties:
$X\bigcirc Y$ iff $X \cap Y = \emptyset$ or $X\subseteq Y$ or $Y\subseteq X$.
Although this is rather basic, it is ...
- 1,882
7
votes
2
answers
957
views
Can models of set theory contain extra ordinals?
In the paper "Complete topoi representing models of set theory" by Blass and Scedrov, they consider a general notion of Boolean-valued model of set theory, and one of the conditions they impose is ...
- 66.8k