2
votes
1answer
89 views

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 ...
2
votes
1answer
58 views

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. ...
5
votes
1answer
152 views

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 ...
2
votes
0answers
64 views

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 ...
5
votes
1answer
102 views

Independent families versus generators in boolean algebras

Let $\kappa$ be an infinite cardinal. A family $\mathcal{A} \subseteq \mathcal{P}(\kappa)$ is independent if, for all $A_1,\ldots,A_n\in\mathcal{A}$ and $i_1,\ldots,i_n\in\{0,1\}$, we have $$ ...
2
votes
0answers
126 views

Basis of a Boolean Algebra

I have a construct that I proved forms a (finite) Boolean Algebra of sets over a given universe. My questions are as follows: Do I immediately know that there exists a unique basis for it? If yes, ...
10
votes
1answer
207 views

Stone's Representation Theorem and The Compactness Theorem

If you're working on $\mathsf {ZF}$ and you assume the compactness theorem for propositional logic, then you have the prime ideal theorem, and thus you can show that the dual of the category of ...
1
vote
1answer
134 views

Free algebra (Boolean algebra)

Could someone give me a simple explanation of Free Algebra on $\kappa$. How to construct free($\omega$). here is it says http://en.wikipedia.org/wiki/Free_Boolean_algebra free($\omega$) is equal to ...
2
votes
1answer
155 views

Infitive distributive law in boolean valued models

I'm posting the problem 2.14 and 2.15 of the book "Set theory" of J.L. Bell. These problem are proposed after the forcing relation chapter and I'm new in this kind of stuff, so I have some little ...
1
vote
1answer
131 views

Question on a mapping between a Boolean algebra and an algebra of sets

On page 81, Set Theory, Jech(2006), to prove the Stone's Representation Theorem, a mapping $\pi$ is defined as Let $B$ be a Boolean algebra. We let $$S=\{p:p \text{ is an ultrafilter on }B\}.$$ ...
1
vote
1answer
168 views

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 ...
0
votes
1answer
166 views

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 ...
10
votes
1answer
142 views

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 $0$ ...
3
votes
1answer
306 views

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?
1
vote
1answer
73 views

Maximal set of pairwise disjoint elements of a dense subset.

Let $B$ be a complete Boolean algebra, and suppose $D \subseteq B$ is a dense subset. How is it possible to construct a maximal set of pairwise disjoint elements of $D$? Is it true that $\sum S = \sum ...
1
vote
0answers
61 views

$\Delta_0$-formulas in the Boolean-valued model $V^B$

I want to show that $\Vert \check x = \check y \Vert = 1$ implies $x = y$, where $\check x$ is the canonical name for $x$ in $V^B$. I'd like try induction over the ranks of $\check x$ and $\check y$ ...
2
votes
1answer
124 views

Question about Cuts in Boolean Algebras

Let $A$ be a Boolean algebra, and let $A^+$ denote the set of non-zero elements of $A$. A cut $U \subseteq A^+$ is a set such that if $q\in U$, then $p\le q \implies p \in U$, for all $p \in A^+$. A ...
3
votes
2answers
102 views

Countable saturation impossible in a complete Boolean algebra?

Let $B$ be an infinite, complete Boolean algebra, and let $\kappa = \operatorname{sat}(B)$. I would like to show that $\kappa$ is uncountable. If we suppose $\kappa$ is countable, that is to say ...
2
votes
2answers
73 views

Is it true that a dense subset of a complete Boolean algebra has supremum 1?

Let $S\subseteq B$ be a dense subset of a complete Boolean algebra $B$. Is is true that $\sum S = 1$? Jech seems to use this fact several times in his book (e.g. the proof of 7.15) but I have been ...
0
votes
1answer
143 views

A Distributivity Law in Complete Boolean Algebras

Let $B$ be a complete Boolean algebra. Define 3 subsets of B as follows: $B_I:= \{ u_{0,i} \mid i \in I \}$ $B_J := \{ u_{1,j} \mid j \in J \}$ $B_{I \times J} := \{ u_{0,i} \cdot u_{1,j} | (i,j) ...
2
votes
1answer
130 views

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 ...
1
vote
1answer
132 views

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 ...
3
votes
3answers
278 views

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 $B$, ...
4
votes
1answer
192 views

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 $x \in B$ ...
1
vote
0answers
182 views

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 ...
6
votes
2answers
198 views

Boolean algebras without atoms

Why is the theory of Boolean algebras without atoms $\omega$-categoric?
1
vote
1answer
839 views

identity and inverse/complement elements in a boolean algebra

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 ...