# Tagged Questions

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### the quotient boolean algebra of $P(\kappa)$ over the nonstationary ideal

Let $\kappa$ be a regular cardinal. Then the quotient boolean algebra over the nonstationary ideal, $P(\kappa)/I_{NS}$ is $\kappa^+$-complete. Specifically, any $S \subseteq P(\kappa)/I_{NS}$ of ...
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### On proving that $\mathcal{P}(\omega)/Finite$ is atomless

As I mentioned elsewhere, I'm working on Schimmerling's A Course on Set Theory. One of the nice features of the book (for me, anyway) is the addition of some interesting exercises on Boolean algebras. ...
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### Cantor-Bernstein theorem for $\sigma$-complete Boolean algebras.

I am working on problem 7.28 from Jech's Set Theory: Let A and B be σ-complete Boolean algebras. Let a and b be elements of A and B respectively. If A is isomorphic to B$\upharpoonright$b and B is ...
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### Fields of sets in which, if the l.u.b. of a subset exists at all, it is the union of the subset

I am learning about boolean algebras and how they can be represented as fields of sets. Stone's representation theorem tells us that every boolean algebra is isomorphic to a field of sets. Consider an ...
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### Problem 1.30 (The Maximum Principle is equivalent to the axiom of choice) (J.Bell, Boolean-valued Models and Independence Proofs, 3rd edition)

Problem 1.30 (The Maximum Principle is equivalent to the axiom of choice) (i) Let $\{a_i : i ∈ I\} ⊆ B$ satisfy $\bigvee_{i∈I} a_i = 1$. A partition of unity $\{b_i : i ∈ I\}$ in B is called a ...
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### Three questions on chapter 7 of Jech's Set Theory

In the proof of Pospisil's Theorem (theorem 7.6) that there are $2^{2^\kappa}$ uniform ultrafilters on $\kappa \geq \omega$, the author writes : Let $\mathcal{A}$ be an independent family of subsets ...
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### Saturated Boolean algebras in terms of model theory and in terms of partitions

Let $\kappa$ be an infinite cardinal. A Boolean algebra $\mathbb{B}$ is said to be $\kappa$-saturated if there is no partition (i.e., collection of elements of $\mathbb{B}$ whose pairwise meet is $0$ ...
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### Cardinality of the set of ultrafilters on an infinite Boolean algebra

Let $\mathfrak B$ be a Boolean algebra with an infinite power $\kappa$. My question is how many ultrafilters does it have? $\kappa$ or $2^\kappa$? Or even smaller?
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### If $B$ is an infinite complete Boolean algebra, then its saturation is a regular uncountable cardinal

I am trying to understand the proof of the statement (Jech 7.15) If $B$ is an infinite complete Boolean algebra, then $\operatorname{sat}(B)$ is a regular uncountable cardinal. I understand the ...
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### A complete Boolean algebra $B$ satisfies the $\kappa$-chain condition if and only if $B$ is $\kappa$-saturated

Let $B$ be a Boolean algebra. Then we say $B$ is $\kappa$-saturated if there is no partition $W$ of $B$ such that $|W| = \kappa$. We say that $B$ satisfies the $\kappa$-chain condition if there is no ...
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### The completion of a Boolean algebra is unique up to isomorphism

Jech defines a completion of a Boolean algebra $B$ to be a complete Boolean algebra $C$ such that $B$ is a dense subalgebra of $C$. I am trying to prove that given two completions $C$ and $D$ of $B$, ...
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### Question about Boolean algebra and ultrafilters

In the following $B$ denotes a Boolean algebra and $\bar{x}$ is the complement of $x$. In my notes there is the following theorem: If $U \subset B$ is an ultrafilter on $B$ then for every $x \in B$ ...
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### Can the reduced product construction generate boolean-valued models?

In model theory, the reduced product construction contains a collection of structures or models, a set I that indexes the collection, and a filter U on I. Ultraproducts are a special case of reduced ...
Why is the theory of Boolean algebras without atoms $\omega$-categoric?
In a boolean algebra, 0 (the lattice's bottom) is the identity element for the join operation $\lor$, and 1 (the lattice's top) is the identity element for the meet operation $\land$. For an element ...