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13
votes
0answers
521 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 ...
11
votes
4answers
898 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 ...
9
votes
3answers
720 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 ...
9
votes
1answer
181 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 ...
9
votes
0answers
387 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 < ...
8
votes
6answers
1k 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 ...
8
votes
2answers
322 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 ...
8
votes
1answer
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 ...
8
votes
0answers
459 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 ...
7
votes
3answers
415 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 ...
7
votes
1answer
446 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 ...
7
votes
2answers
786 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 ...
7
votes
1answer
260 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 ...
7
votes
2answers
434 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 ...
7
votes
1answer
256 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 ...
6
votes
2answers
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 ...
6
votes
4answers
709 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 ...
6
votes
2answers
465 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 ...
6
votes
1answer
362 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 ...
6
votes
0answers
225 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 ...
5
votes
3answers
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 ...
5
votes
4answers
340 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) ...
5
votes
2answers
649 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 ...
5
votes
2answers
350 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 ...
5
votes
2answers
172 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
1answer
189 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
2answers
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 ...
4
votes
2answers
801 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
2answers
218 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: ...
4
votes
2answers
591 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) = ...
4
votes
1answer
300 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 ...
4
votes
1answer
192 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 ...
4
votes
1answer
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
2answers
333 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 ...
3
votes
2answers
278 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 $I=J_{bd}$ ...
3
votes
2answers
311 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 ...
3
votes
1answer
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 ...
3
votes
2answers
182 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 ...
3
votes
1answer
109 views

Systems of equations in Boolean Algebra

I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by ...
3
votes
1answer
92 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 ...
3
votes
2answers
339 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 ...
3
votes
1answer
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 ...
3
votes
1answer
724 views

A possible generalization of the Cox Theorems (boolean algebra => bayesian probabilities)

This post is an attempt to gather people to solve a particular problem in mathematics, something that can actually be published and seems to me simple enough to test this mathoverflow as a ...
3
votes
1answer
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 ...
3
votes
2answers
105 views

About a construction of Borel $\sigma$-algebra associated to a lattice

Let $(\mathcal{A}, \cup, \cap)$ a lattice (with minimum and maximum elements $\bot$ and $\top$). Let $X\subset \mathcal{A}$ a generator set (a set of minimal cardinality that generate $\mathcal{A}$ ...
2
votes
2answers
328 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 ...
2
votes
4answers
498 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 ...
2
votes
2answers
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 ...
2
votes
1answer
420 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 ...
2
votes
1answer
149 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?