All Questions
Tagged with set-theory boolean-algebras
80 questions
5
votes
1
answer
148
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
9
votes
1
answer
252
views
Copy of $P(\omega)/\mathrm{fin}$ on $\omega_1$
$\newcommand{\fin}{\mathrm{fin}}$Under what hypotheses does there exist a uniform ideal $I$ on $\omega_1$ such that $P(\omega_1)/I \cong P(\omega)/\fin$? What is the consistency strength?
It follows ...
- 18.6k
4
votes
0
answers
139
views
Commutativity of a diagram between complete embeddings
Suppose $\mathbb{P}_0$, $\mathbb{P}_1$ and $\mathbb{P}_2$ are separative posets such that $\mathbb{P}_2$ projects into $\mathbb{P}_1$ and $\mathbb{P}_1$ projects into $\mathbb{P}_0$, i.e. there are ...
- 533
6
votes
1
answer
113
views
Projections between complete boolean algebras
Let $P$ and $Q$ be complete boolean algebras. Suppose that $\dot H$ is a $P$-name such that $1_P\Vdash\dot H$ is $Q$-generic. For each $p\in P$, let $A_p$ be the set of $q\in Q$ such that $p\Vdash q\...
- 533
6
votes
1
answer
220
views
On the number of complete Boolean algebras
In their 1972 paper On the number of complete Boolean algebras Monk and Solovay showed that if $\lambda$ is an infinite cardinal, then there are $2^{2^\lambda}$ many isomorphism types of
complete ...
- 32.1k
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
11
votes
3
answers
792
views
When are two forcing posets "the same"?
Let $B$ and $C$ be complete Boolean algebras. To avoid triviality I may also want them to be atomless. For $b\in B$ nonzero, denote $B\upharpoonright b=\{p\in B:p\leq b\}$, which can be viewed as a ...
- 877
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
11
votes
1
answer
471
views
A strictly descending chain of subalgebras of $P(\omega)/_{\mathrm{fin}}$
Consider the algebra $B=P(\omega)/_{\mathrm{fin}}$ (the quotient of the power set of natural numbers modulo the ideal of finite sets). Is there an infinite strictly descending chain $\{A_i\mid i\in I\}...
- 1,112
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
2
votes
2
answers
176
views
Why is a Boolean algebra being $\kappa$-saturated upward closed in $\kappa$?
A Boolean algebra $B$ is defined (e.g. in Jech) to be $\kappa$-saturated if there is no partition $W$ of $B$ where $|W|=\kappa$. He seems to assume that this implies $|W|<\kappa$ for any partition ...
- 1,219
3
votes
0
answers
152
views
Boolean Algebra of size $2^{<\kappa}$ without an $\aleph_1$-complete ultrafilter
For this post we work only with cardinals that live below the first measurable.
Assume that $\kappa$ is singular and $\kappa<2^{<\kappa}<2^\kappa$.
Question: Is it possible to have a Boolean ...
- 2,882
5
votes
2
answers
314
views
Self-homeomorphism of Stone-Čech boundary with an isolated fixed point
$\DeclareMathOperator\bso{\beta^*\!\omega}\DeclareMathOperator\Homeo{Homeo}$Let $\bso$ be the complement of the countable discrete space $\omega$ in its Stone-Čech compactification $\beta\omega$ (some ...
- 63.9k
12
votes
0
answers
406
views
An internal notion of freeness for complete Boolean algebras
Background and Definition
Gaifman and Hales showed that there are no infinite free complete Boolean algebras.
But let a complete Boolean algebra $B$ be internally free if there is a set $X\subseteq B$ ...
- 121
6
votes
0
answers
166
views
Rigid boolean inclusions?
A boolean algebra $B$ is rigid if it has no nontrivial automorphisms and atomless if it has no minimal nonzero elements. $A \subseteq B$ is a complete boolean inclusion if $B$ is complete and $A$ is a ...
- 271
6
votes
1
answer
337
views
Boolean valued models in a general setting
It is well known that Boolean valued models play significant roles for set-theoretic purposes. But how well-studied are Boolean valued models in a more general setting, as models for random first-...
7
votes
1
answer
397
views
A set theoretic question arising from trying to understand a sheaf cohomology question
I'm trying to understand the footnote to Example 5.3 in Wiegand - Sheaf cohomology of locally compact totally disconnected spaces which is about constructing a locally compact Hausdorff and totally ...
- 38.6k
8
votes
1
answer
480
views
Intuition behind Boolean-valued models of set theory
$\DeclareMathOperator\Card{Card}$The book Forcing Eine Einführung in die Mathematik der Unabhängigkeitsbeweise by Hoffmann provides an intuition behind boolean valued models of set theory which I will ...
- 123
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
6
votes
1
answer
280
views
Reduced products of complete Boolean algebras
I expect that complete Boolean algebras are not closed under reduced products modulo $\kappa$-complete filters, for any regular cardinal $\kappa$. Is it true? And, is a reference for this?
- 2,160
10
votes
1
answer
354
views
Elementary equivalence between $n\mapsto n+1$ and its inverse on the Stone-Čech remainder?
Consider structures $(A,f)$ encoding a Boolean algebra $A$ endowed with an automorphism $f$. There is an obvious notion of isomorphism between such structures.
Consider the endomorphism $\hat{\Phi}$ ...
- 63.9k
14
votes
2
answers
502
views
Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?
Let $\beta\omega$ be the Stone-Čech compactification of the discrete infinite countable space $\omega$, and $\beta^*\omega=\beta\omega\smallsetminus \omega$ is the Stone-Čech remainder.
The map $j:n\...
- 63.9k
5
votes
1
answer
371
views
A problem of non-emptiness of intersections of certain chains of regular open sets
Let $X$ be a topological space and $\mathrm{RO}(X)$ its complete boolean algebra of regular opens. Define well inside relation: $$U\prec V\iff\overline{U}\subseteq V.$$ Let $\mathcal C\subseteq\mathrm{...
- 1,112
7
votes
2
answers
236
views
Linear suborders of $(P(\omega),\subseteq)$
Consider the partial order $(P(\omega),\subseteq)$. Let $L$ be a dense linear suborder. Does $L$ have a countable dense subset?
(Note that it contains a copy of $\mathbb R$, via Dedekind cuts of $\...
- 18.6k
4
votes
1
answer
270
views
Infinite distributive laws in atomless free sigma-algebra
Let $\frak{A}$ be the free $\sigma$-algebra on $\omega_1$ free $\sigma$-generators. Then $\frak{A}$ is not completely distributive because it is atomless. However, is it $\omega$-distributive in the ...
- 207
6
votes
1
answer
255
views
Finite covers of Boolean algebras by their subalgebras
It is a student exercise that no group can be represented as a set-theoretic union of its two proper subgroups. The same also can be shown for Boolean algebras. On the other hand, it's not hard to ...
- 1,151
2
votes
1
answer
193
views
Semi-rigid boolean algebras
A boolean algebra is rigid if it has no nontrivial automorphisms. Call it semi-rigid if none of its nontrivial automorphisms has any fixed points other than 0 and 1.* The four-element algebra $\{0, b, ...
- 271
8
votes
4
answers
714
views
Are there $2^{\aleph_0}$ pairwise non-isomorphic Boolean algebra structures on $\omega$?
Is there a collection of $2^{\aleph_0}$ pairwise non-isomorphic countable Boolean algebras?
Equivalently, are there $2^{\aleph_0}$ pairwise non-homeomorphic closed subsets in the Cantor space?
- 48.9k
2
votes
0
answers
240
views
3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators
During my studies, I came across several different Stone spaces, e.g.:
(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;
...
- 207
2
votes
0
answers
147
views
C.c.-ness of a forcing notion based on an atomless complete Boolean algebra
Given $\mathbb{B} = \langle B, \wedge, \vee, \neg, 0, 1 \rangle$ an atomless complete Boolean algebra that has a $< \mkern-4mu \kappa$-closed dense subset and is $\kappa^+$-c.c., we define a ...
- 1,328
2
votes
2
answers
279
views
About the existence of a particular kind of "splitting" function on atomless complete Boolean algebras
Let $\mathbb{B} = \langle B, \wedge, \vee, \leq, \neg, 0, 1 \rangle$ be an atomless complete Boolean algebra.
We call $f$ a splitting function on $\mathbb{B}$ iff
$f : B-\{1\} \longrightarrow B \...
- 1,328
8
votes
5
answers
464
views
Ideals on $\mathbb N$ and large sets that have small intersection
Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable:
$$A,...
- 11.3k
6
votes
2
answers
482
views
Complete atomless Boolean algebras with abelian automorphism group
Is there any example of a complete atomless Boolean algebra with a non-trivial abelian automorphism group?
This is equivalent, by Stone duality, to asking for an extremally disconnected compact ...
- 3,115
9
votes
1
answer
385
views
Embeddings of Boolean algebras in $\wp(\omega)/Fin$
If we assume MA+¬CH, then every boolean algebra with cardinality smaller than the continuum embeds in ℘(ω)/Fin. A proof of this result can be found in Theorem 1.1, Chapter 8 of the book "Hausdorff ...
- 419
11
votes
1
answer
2k
views
Is every complete Boolean algebra isomorphic to the quotient of a powerset algebra?
Is every complete Boolean algebra isomorphic to a quotient, as a Boolean algebra, of some powerset algebra $\wp(X)$?
It is not true for arbitrary Boolean algebras, see the comments, or see my MathSE ...
- 263
6
votes
1
answer
606
views
Boolean ultrapower of V[G] by G
In Joel David Hamkins's "Well-founded Boolean Ultrapowers as Large Cardinal Embeddings", it is mentioned that if $U \in \mathbf{V}$ is an ultrafilter of a complete Boolean algebra $\mathbb{B}$ and $U$ ...
- 1,328
9
votes
0
answers
356
views
Direct limits of $\sigma$-centered forcing notions
It is quite well known that
Any FS (finite support) iteration of length $<\mathfrak{c}^+$ of $\sigma$-centered posets is $\sigma$-centered (see e.g. here).
Now consider the following question: ...
- 791
8
votes
2
answers
289
views
Does $\aleph_0$-density of regular open algebra entail existence of countable basis?
Suppose that the family $\mathrm{RO}(X)$ of regular open subsets of $(X,\mathscr{O})$ is a basis of $X$. Let the density of $\mathrm{RO}(X)$ (considered as Boolean algebra) be $\aleph_0$.
Does $X$ ...
- 1,112
3
votes
1
answer
436
views
Stone topological Boolean algebras
I am looking for an initial reference for a theorem which is known, namely:
Theorem: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological ...
- 774
11
votes
1
answer
707
views
partitions of Boolean algebras
A partition of a Boolean algebra is a collection of pairwise disjoint nonzero elements with supremum 1. For any infinite Boolean algebra $A$ let $a(A)$ be the least size of an infinite partition of $A$...
- 883
12
votes
2
answers
643
views
A curiosity on complete homomorphisms of boolean algebras
The question may be trivial, but has eluded me, may be it is more appropriate for mathstack-exchange.
Let $B$, $C$ be boolean algebras and $i:B\to C$ be an homomorphism.
By Stone duality to each such ...
- 489
6
votes
1
answer
378
views
Weak equivalence over forcing notions
We know there are several definitions about forcing equivalence which imply that two forcings notions can be equivalent or not. In general we like to know the similarity between generic extensions by ...
- 2,381
5
votes
2
answers
555
views
How "strong" is the existence of a non trivial ultrafilter on $\omega$?
Obviously the question in the title alone doesn't make sense so I'll develop on the context and then I'll ask my question :
Studying $AD$ (axiom of determinacy) I had to prove that $AD$ and $AC$ are ...
- 180
1
vote
3
answers
290
views
Is every c.c.c. non-atomic partial order of size $\omega_1$ a union of countable complete suborders?
We say that $\mathbb{P}$ is a complete suborder of $\mathbb{Q}$, if it is a suborder, and maximal antichains in $\mathbb{P}$ remain maximal antichains in $\mathbb{Q}$
As the title says, is every c.c....
- 193
8
votes
1
answer
294
views
Boolean-Valued Models: Why is $\| x=y \| \cdot \| \phi(x) \| \leq \| \phi(y) \|$?
Let $B$ be a complete Boolean algebra. Jech defines a Boolean-valued model $\mathfrak{A}$ of the language of set theory to consist of a Boolean universe $A$ and functions of two variables with values ...
user83393
2
votes
1
answer
307
views
Boolean algebras and free filters generated by chains
Suppose $\mathbf{B}$ is a complete Boolean algebra with an infinite domain $B$. Suppose $\mathbf{B}$ is atomic (i.e. every element is the supremum of some set of atoms). This algebra contains the co-...
- 1,112
7
votes
0
answers
182
views
rigidity of $\mathcal P(\omega_1) / NS$ under MA
In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it ...
- 18.6k
4
votes
1
answer
352
views
Cohen algebra and $\mathcal P(\omega)/ \mathrm{fin}$
Let $C$ be the Cohen algebra, the boolean completion of the partial order of finite partial functions from $\omega$ to 2, ordered by reverse inclusion. Does there exist an ideal $I$ on $C$ such that $...
- 18.6k
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 $\...
- 2,231