The boolean-algebras tag has no usage guidance.

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442 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 ...

**1**

vote

**1**answer

105 views

### distributive sublattices of atomistic ortholattices

Let $L$ be an atomistic ortholattice (i.e. every element can be written as a join of atoms) with top and bottom elements 0 and 1, and let $M$ be a distributive atomic sub-ortholattice of $L$.
Is $M$ ...

**1**

vote

**3**answers

338 views

### Why the preimage rather than image in Stone-type dualities.

I am seeking a deeper understanding of the representation of set-based objects in terms of Boolean algebras.
Let $\wp(A)$ be the set of subsets of a set $A$. A relation $R \subseteq A \times B$ ...

**1**

vote

**1**answer

154 views

### What can we infer about the size of a complete Boolen algebra, given it is $\kappa$-c.c.?

More specifically, if we only know that a complete Boolean algebra, $\mathbf{B}$, is $\kappa$-c.c., can we give a (reasonably tight) upper bound to the size of $\mathbf{B}$ in terms of $\kappa$?
...

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votes

**1**answer

116 views

### perfect space without convergent long sequences

Is there a boolean space $X$ without isolated points with the property that no point $x\in X$ is the limit of a long sequence $(x_i)_{i\in I}$ from $X\setminus \lbrace x\rbrace $ ('long sequence' here ...

**1**

vote

**3**answers

274 views

### Cardinality of the set of maximal ideals in a Boolean ring/algebra

If B is a Boolean ring is of uncountable cardinality c, does B have 2^c distinct maximal ideals ?
Can you please give me a reference where this question is answered (hopefully) positively ? Thanks

**5**

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**2**answers

178 views

### Quotients of Cantor cubes onto spaces

Let $\lambda$ be an infinite cardinal. Consider the Cantor cube $\Delta_\lambda = \{0,1\}^\lambda$. It is a standard fact in topology that the topological weight (= minimal cardinality for a basis) of ...

**5**

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**2**answers

365 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 ...

**3**

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**1**answer

150 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 ...

**2**

votes

**1**answer

134 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 $B$?
...

**6**

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**1**answer

205 views

### On intermediate transitive models for ZFC between M an M[G]

Let $P$ be a forcing notion. Let $B(P)$ be the boolean completion of $P$ and $i : P \rightarrow B(P)$ be the corresponding dense embedding (in $B(P)^{+}$). Let $G$ be $B(P)$-generic over $M$, the ...

**4**

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**1**answer

200 views

### Categorical properties of metric Boolean algebras

According to Kolmogorov ("Algèbres de Boole métriques complètes", VI Zjazd Matematykòw Polskich, 1948, english translation Phil. Studies, 1995, 77, 57-66), a Boolean algebra $(B, \wedge, \vee,-,1,0)$ ...

**2**

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**2**answers

364 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 ...

**4**

votes

**1**answer

312 views

### Continuous Strictly Positive Measures on Countable Boolean Algebras

This is a followup to:
Strictly Positive Measures on Countable Boolean Algebras
Suppose a countable Boolean algebra B is a subalgebra of the power set of the reals. (For example, let B be the ...

**7**

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**1**answer

272 views

### Strictly Positive Measures on Countable Boolean Algebras

Let $B$ be a Boolean Algebra.
A strictly positive measure on $B$ is a function $m$ from $B$ to $[0,1]$ such that (i) $m(b)=0$ iff $b=0$, (ii) $m(1)=1$, and (iii) $m(a+b)=m(a)+m(b)$ whenever $a$ and ...

**5**

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**3**answers

369 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 ...

**-1**

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**1**answer

207 views

### Simplifying Boolean Algebra [closed]

I have the following boolean algebra equation:
(A + B + C)(D + ~C)
I don't know if my equation is in standard form, so just to be explicit, by '+' I mean OR and the parenthesis mean AND. ~ is NOT.
...

**5**

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**2**answers

676 views

### 0-dimensional locally compact space

What is an example of a 0-dimensional locally compact Hausdorff space X for which the Cech-Stone compactification beta(X) is NOT 0-dimensional?
It is known that if X is a 0-dimensional locally ...

**2**

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**2**answers

235 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 ...

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**2**answers

1k views

### Examples for “nice” Boolean algebras that are not complete or not atomic

A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms).
Boolean Algebras that are complete as ...

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**2**answers

220 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: ...

**2**

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**1**answer

194 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 ...

**3**

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**2**answers

319 views

### ED compact $K$ such that $C(K)$ is not a dual Banach space

Almost every mathematician working in von Neumann algebras says that there are certainly extremely disconnected compact spaces $K$ such that $C(K)$ is not a dual space (they are not hyperstonean). ...

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votes

**1**answer

340 views

### Extending a complete lattice to get a “nice” Boolean lattice

Suppose we have a complete lattice. Which additional axioms (e.g. distributivity axioms) are needed to obtain a Boolean lattice in which complement(a) = lub{b | b /\ a = bottom} = glb{b | b / a = ...

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188 views

### Chain of ideals in a BA

Suppose $\mathfrak{A}$ is a Boolean algebra and $\mathfrak{J}$ is chain of ideals in $\mathfrak{A}$ ordered by inclusion such that none of its elements is countably generated. Clearly, the union ...

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**2**answers

490 views

### Is there something like a Heyting Ring?

I would like to know whether a Heyting algebra gives rise to ring in a similar way that a Boolean algebra gives rise to a Boolean ring. In a Boolean algebra $(B,\lor,\land,\lnot,0,1)$ I can define ...

**0**

votes

**1**answer

215 views

### Low degree approximation of the polynomial extension of the logical-or function

Let $x\in\{0,1\}^n$ be a binary vector of dimension $n$, and let $OR(x)$ be the "logical or" function (i.e., returns $1$ if at least one of the coordinates is $1$ and otherwise returns $0$).
Consider ...

**6**

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**0**answers

233 views

### Counting Copies of a BA within a BA: Arbitrarily Many versus 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 ...

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345 views

### Antichains and measure-preserving actions on Boolean algebras

Let $G$ be a group of automorphisms of the countable atomless Boolean algebra $B$. Suppose that every orbit of $G$ on $B$ is an antichain. Does it follow that $G$ preserves a non-zero (probability) ...

**2**

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**2**answers

296 views

### “Duals” of Lindenbaum algebras

From Wikipedia I learn:
The Lindenbaum algebra A of a theory T consists of the equivalence
classes of sentences of T. The operations in A are inherited from those in T.
If there are ...

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**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 ...

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**3**answers

761 views

### A unique ultrafilter extending a union of filters?

Original Question:
Let $\mathcal{P}(\omega)/fin$ denote the Boolean algebra formed from $\mathcal{P}(\omega)$ by modding out by the ideal $fin$ of finite subsets of $\omega$. As a first pass at the ...

**4**

votes

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634 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) = ...

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**0**answers

616 views

### Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?

Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra.
Moreover, a morphism of commutative von Neumann algebras induces
a continuous morphism of the corresponding ...

**2**

votes

**1**answer

450 views

### Free product of Boolean algebras

Given a family of Boolean algebras $\mathcal{B}=\{B_i\colon i\in I\}$ with respective Stone spaces $S_i$. Recall that the algebra of clopen (both closed and open) subsets of the product space
...

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**0**answers

215 views

### Defining filters in closure algebras: reference request

A closure algebra C is a boolean algebra B together with a unary closure operator, and additional axioms, the Kuratowski axioms, that the closure operator must satisfy. (The Wikipedia article prefers ...

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**4**answers

954 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 ...

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votes

**1**answer

376 views

### Is it possible to define a closure operator in terms of partial ordering?

For boolean algebra, let's take Roman Sikorski's Boolean Algebras as our reference. After giving a set of axioms, he proves (p.9) that the join of A and B is the least element of the algebra such that ...

**8**

votes

**1**answer

483 views

### Coproducts of complete Boolean algebras

Does the category of complete Boolean algebras have binary coproducts?
Note that this category does not have countable coproducts. Indeed, the coproduct of countably many copies of the four element ...

**1**

vote

**5**answers

610 views

### Is there such a thing as the sigma-completion of a Boolean algebra?

Hi all,
Suppose that $\mathcal{B}$ is a Boolean algebra. It there a way to extend $\mathcal{B}$ to a smallest Boolean algebra $\mathcal{B}'$ that contains an isomorphic copy of $\mathcal{B}$ and is ...

**6**

votes

**4**answers

820 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 ...

**7**

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**2**answers

809 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 ...

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**2**answers

327 views

### Semilattices in atomless boolean algebras

Let S be a bounded semilattice without maximal elements. Can we always construct an atomless boolean algebra B, containing S as a subsemilattice, such that S is cofinal in B-{1}? That is, for every ...

**8**

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**6**answers

2k views

### Generalizations of Boolean posets/lattices

A Boolean lattice has a number of rather nice properties which give it a central role in many parts of combinatorics. For instance, it's a lattice, it can be augmented with a ring structure, it can ...