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Results tagged with boolean-algebras
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user 11145
A Boolean algebra is a commutative ring satisfying x²=x for every x, and sometimes required to have a unit; they have characteristic 2. For coding theory (notably dealing with subsets linear subspaces of spaces of Boolean functions), rather use the [coding-theory] or [linear-algebra] tag.
4
votes
2
answers
273
views
extending $\sigma$-complete boolean homomorphism
I'm not sure if this is research level, so feel free to vote to migrate.
Suppose we have a complete boolean algebra $A$, with a dense, $\sigma$-complete subalgebra $B$, and a $\sigma$-complete homomo …
- 18.7k
4
votes
2
answers
278
views
Quasi-dense subsets of boolean algebras
Definition: Let $B$ be a boolean algebra. Say $X \subseteq B$ is quasi-dense in $B$ if for all $b \in B$, there is $x \in X \setminus$ { $0,1$ } such that either $x \leq b$ or $b \leq x$.
Question: …
- 18.7k
12
votes
Accepted
Independent families versus generators
I think the answer is always no. Consider the algebra $\mathcal{B}$ generated by the independent family and the ideal $J$ of subsets of $\kappa$ of size $<\kappa$. Then look at $\mathcal{C} = \{ X \s …
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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 $\ …
- 18.7k
4
votes
Accepted
Boolean completion (of a forcing notion) isomorphic to each of its cones
Your second statement is correct, simply because boolean completions are unique up to isomorphism.
For the stronger statement, let $b \in \mathbb{B}^+$. Let $\{ p_\alpha : \alpha < \kappa \}$ be a m …
- 18.7k
8
votes
centeredness in forcing iterations
It's consistent that the answer is no. Bartoszynski and Judah prove the following on page 26 in their book, Set Theory: On the Structure of the Real Line:
Assume $MA_\kappa$. Then a partial orde …
- 18.7k
6
votes
Accepted
How complete is $\mathcal P(\kappa)/J_{bd}$?
If $\kappa$ is regular, then $\mathcal{P}(\kappa)/J_{bd}$ is a $\kappa$-complete boolean algebra. If $\langle A_\alpha : \alpha < \delta < \kappa \rangle$ is a sequence of subsets of $\kappa$, then t …
- 18.7k
2
votes
subalgebra of a simple forcing
I have answers the original questions, by way of counterexample, which I will sketch. The "new related question" remains unsolved. I am grateful to Mohammad Golshani for his answer to this question, …
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7
votes
2
answers
489
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 $\ …
- 18.7k
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) = $\alep …
- 18.7k
4
votes
Differences between the reduced Borel field and the category algebra of a space
There is no difference, since every set with the Baire property is equivalent to an open set (thus Borel set) modulo the meager ideal.
- 18.7k
7
votes
2
answers
468
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 m …
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4
votes
1
answer
345
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 …
- 18.7k
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 h …
- 18.7k
2
votes
Suslin algebras
Jech, Thomas J.
Some combinatorial problems concerning uncountable cardinals.
Ann. Math. Logic 5 (1972/73), 165–198.
Section 5 contains the forcing for arbitrarily big Suslin algebras. See also:
…
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