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1answer
104 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
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3answers
334 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
1answer
150 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
1answer
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
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3answers
273 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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2answers
176 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
2answers
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
votes
1answer
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
1answer
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
votes
1answer
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
votes
1answer
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
votes
2answers
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
1answer
311 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
1answer
269 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
3answers
368 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 ...
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1answer
206 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
2answers
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
votes
2answers
234 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
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2answers
967 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
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
votes
1answer
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
votes
2answers
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). ...
2
votes
1answer
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 = ...
3
votes
2answers
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 ...
6
votes
2answers
486 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
1answer
209 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
votes
0answers
229 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
4answers
344 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
votes
2answers
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 ...
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 ...
9
votes
3answers
752 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
2answers
630 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) = ...
15
votes
0answers
609 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
1answer
448 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 ...
1
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0answers
214 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 ...
11
votes
4answers
949 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 ...
6
votes
1answer
375 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 ...
7
votes
1answer
473 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
5answers
607 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
4answers
809 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
votes
2answers
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 ...
3
votes
2answers
325 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
votes
6answers
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 ...