The boolean-algebras tag has no wiki summary.

**7**

votes

**1**answer

228 views

### On $V$-decisive and weakly homogeneous forcings

Suppose that $\Bbb P$ is a forcing in $V$, we say that $\Bbb P$ is $V$-decisive if whenever $\varphi(x_1,\ldots,x_n)$ is a statement in the language of forcing, and $u_1,\ldots,u_n\in V$ then $1_{\Bbb ...

**0**

votes

**0**answers

26 views

### Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$.
I am able to write each event as a sum of distinct events that form a partition of the space.
My goal is to find all the ...

**3**

votes

**1**answer

91 views

### Boolean completion (of a forcing notion) isomorphic to each of its cones

Suppose $ \mathbb{P} := (P, {\leq_P}, 1_P) $ is a separative partial order. Let $ \mathbb{B} := \operatorname{RO}(\mathbb{P}) $ denote the Boolean completion.
Fix some dense embedding $ i \colon P ...

**-4**

votes

**1**answer

66 views

### How to find matrix representations of a boolean algebra? [closed]

Given a boolean algebra with a finite number of elements {a, b, c, ...}, and the usual operations: $\cup, \cap, \neg$.
How to find matrix representations of the elements such that:
boolean $\cup$ ...

**1**

vote

**2**answers

221 views

### Question on separability of a measure

Following this question here this question come to mind.
Consider a measured σ-algebra $(S,\mu)$ . Assume that μ is normalized to have total weight 1, and that S is complete (contains all subsets of ...

**1**

vote

**1**answer

88 views

### Krull dimension of dense extensions

Let $A$ be a boolean algebra and let $B\leq A$ be a boolean sub-algebra which is dense (for all $0\neq a\in A$, there is a $0\neq b\in B$ such that $b\leq a$). We suppose also that $B$, as a partially ...

**2**

votes

**1**answer

147 views

### Sigma-complete Lindenbaum algebras? [closed]

Is there any calculus whose algebraization is a sigma-complete Lindenbaum algebra, i.e., a sigma-complete Boolean algebra after identification of equivalent formulas?

**4**

votes

**2**answers

85 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 ...

**3**

votes

**1**answer

82 views

### existence of more distributive Boolean lattices

Are there a Boolean lattice $(X,\le)$ and a nonempty collection $(a_{ij})_{i\in I,j\in J}\subseteq X$ such that
$$\bigvee_{i\in I}\bigwedge_{j\in J}{a_{ij}}\ne \bigwedge_{f:I\to J}\bigvee_{i\in ...

**2**

votes

**1**answer

160 views

### The Universal Algebra of a sigma-Algebra

I am searching for the 'dual' algebraic structure of a Sigma Algebra. The notion of duallity is like on the case of the Boolean Algebra and Set Algera.
If X is a set, the complement and intersection ...

**-1**

votes

**1**answer

156 views

### Algebra generated by a tree [Edit] [closed]

Suppose that $(T,\leq)$ is a partially ordered set, we say $T$ is a tree* if for every $i\in T$, $\{s: s\in T, s\leq t\}$ is a well-founded chain.
What I need to know is: Can the algebra ...

**4**

votes

**1**answer

183 views

### Uncountable atomless subalgebras of the Boolean algebra of all Jordan measurable sets in [0,1]

Definition: Suppose $\mathcal A$ is
the Boolean algebra of all Jordan measurable sets in $I=[0,1]$ (i.e $\mathcal A=\{A\subseteq I: \mu(\partial(A))=0\}$, where $\mu$ is the Lebesgue measure and ...

**0**

votes

**1**answer

120 views

### additive measure on countable algebras

I was wondering, can the following theorem be true for finitely additive measures defined on algebras not $\sigma$-algebras. (Theorem is in Bogachev's Measure Theory Vol I).
I was not sure about ...

**7**

votes

**3**answers

390 views

### Which Sigma-Ideals in a Sigma-Algebra are Ideals of Null Sets?

My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space ...

**0**

votes

**1**answer

141 views

### Stone space of measure algebra [closed]

let $\lambda$ be the Lebesgue measure on the unit interval $I=[0,1]$, and $Leb(I)$ be the Boolean algebra of Lebesgue measurable in $I$ and $\mathcal{N}$ the family of Null sets. The measure algebra ...

**2**

votes

**1**answer

136 views

### Eigenvalues of a matrix constructed with simple logic

If a matrix can be constructed with simple bit-logic operations, is it also possible to find Eigenvalues with logic?
First I'll just say that my knowledge of logic is pretty much limited to ...

**8**

votes

**0**answers

450 views

### Full conditional probabilities and versions of AC?

A probability is a finitely additive measure on a boolean algebra with total measure $1$.
A function $P:\scr B \times (\scr B - \{ 0 \})$ is a full conditional probability on $\scr B$ (for a boolean ...

**8**

votes

**1**answer

296 views

### Nontrivially nontrivial automorphisms of $P(\omega_1)/fin$

Velickovic proved (Theorem 4.1 of OCA and automorphisms of $\mathcal{P}(\omega)/\mathrm{fin}$) that, assuming OCA and $\rm MA_{\aleph_1}$, every (Boolean algebra) automorphism of ...

**1**

vote

**3**answers

193 views

### Complete sets of functions

A (finite) set $S$ of boolean functions is called functionally complete if every boolean function can be presented as a finite composition of functions from $S$. For example, $\{ \neg,\wedge \}$ is ...

**9**

votes

**1**answer

179 views

### Independent families versus generators

I asked this question on M.SE a while ago and got no answers, so I'm asking it here.
Let $\kappa$ be an infinite cardinal. A family $\mathcal{A}\subseteq\mathcal{P}(\kappa)$ is independent if for ...

**0**

votes

**1**answer

217 views

### Algorithm to efficiently sum N boolean numbers. [closed]

I am looking for a fast algorithm to do the following task: Given $N$ numbers $a_i, i=1,..., N$, where $a_i$ can be equal to $0$ or $1$, compute the number $s \equiv \sum_{i=1}^N a_i$ in base 2. ...

**3**

votes

**2**answers

277 views

### How complete is $\mathcal P(\kappa)/J_{bd}$?

While it is true that $\mathcal P(\kappa)$ is a complete Boolean algebra, it is not necessarily true that $\mathcal P(\kappa)/I$ is complete for an ideal $I$. In particular if we consider $I=J_{bd}$ ...

**2**

votes

**2**answers

311 views

### Embedding a brouwerian lattice into a boolean lattice

I have already asked a similar question at
http://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra
but have received no answer.
Sorry, I ask a ...

**9**

votes

**0**answers

384 views

### Existence of a regular subposet which collapses everything except the top cardinal

Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular $\kappa < ...

**1**

vote

**1**answer

137 views

### translating a given boolean function to universal boolean function

A Boolean function U($z_1$, $z_2$ ..... , $z_m$) is universal for given n > 1 if it realizes all Boolean functions f($x_l$ ..... $x_n$) by substituting for each $z_i$ with a variable of the set {0, 1,
...

**3**

votes

**2**answers

326 views

### Cohen algebra (generalization)

Let Bor($X$) = class of all borel subsets of $X$. Cohen algebra is defined as Bor(X) modulo the ideal of meager sets.
The Cohen algebra has a combinatorial : it is the unique atomless complete ...

**1**

vote

**0**answers

113 views

### Can random elements be defined in terms of a measure algebra?

Let $(\Omega,\Sigma,\mu)$ be a probability space, $(X,\mathcal{X})$ be a measurable space and $R(\Omega,X)$ be the set of equivalence classes of measurable functions from $\Omega$ to $X$ under almost ...

**2**

votes

**4**answers

482 views

### Products of Boolean algebras and probability measures thereon

These are really two questions, but the second presupposes the first.
First, let $( B_i )_{i\in I}$ be an arbitrary family of Boolean algebras. I want to directly form a product of them that is like ...

**7**

votes

**2**answers

433 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

101 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

313 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

130 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 $\kappa$?
...

**3**

votes

**1**answer

113 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

239 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

**4**

votes

**2**answers

166 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**

votes

**2**answers

347 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**

votes

**1**answer

144 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

126 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 $B$?
...

**5**

votes

**1**answer

186 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**

votes

**1**answer

186 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 $(B, \wedge, \vee,-,1,0)$ ...

**3**

votes

**2**answers

334 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

295 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**

votes

**1**answer

252 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**

votes

**3**answers

362 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**

votes

**1**answer

199 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**

votes

**2**answers

643 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**

votes

**2**answers

226 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 ...

**4**

votes

**2**answers

756 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 ...

**4**

votes

**2**answers

216 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**

votes

**1**answer

192 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 ...