Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

The tag has no usage guidance.

0
votes
0answers
57 views

Stone Space in Ergodic Theory [on hold]

Any use of Stone space in Ergodic theory? Where I mean for example replace in theorems where you assume Polish space or any other suitable spaces which appear in classical ergodic theory literature, ...
3
votes
1answer
66 views

Ultrafilters preserving infinite joins

A filter $U$ over a boolean algebra (isomorphic to a powerset) $A$ "preserves" a join $a = \bigcup_{i\in I}a_i$, if $a\in U$ implies $a_i\in U$ for some $i\in I$. (A join $a$ is infinite if $I$ is.) ...
2
votes
0answers
115 views

C.c.-ness of a forcing notion based on an atomless complete Boolean algebra

Given $\mathbb{B} = \langle B, \wedge, \vee, \neg, 0, 1 \rangle$ an atomless complete Boolean algebra that has a $< \mkern-4mu \kappa$-closed dense subset and is $\kappa^+$-c.c., we define a ...
1
vote
1answer
112 views

Characterization of state spaces of Boolean algebras

A state space of a Boolean algebra is a Choquet simplex but not all Choquet simplices can be viewed as state spaces of Boolean algebras. Is it known which Choquet simplices are precisely state spaces ...
2
votes
2answers
243 views

About the existence of a particular kind of “splitting” function on atomless complete Boolean algebras

Let $\mathbb{B} = \langle B, \wedge, \vee, \leq, \neg, 0, 1 \rangle$ be an atomless complete Boolean algebra. We call $f$ a splitting function on $\mathbb{B}$ iff $f : B-\{1\} \longrightarrow B \...
7
votes
5answers
375 views

Ideals on $\mathbb N$ and large sets that have small intersection

Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable: $$A,...
1
vote
2answers
77 views

Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$?

Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be regular open if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A ...
1
vote
1answer
109 views

Is the boundary of an open set in a $\sigma$-space empty?

Recall that a Boolean space is a $\sigma$-space in case the closure of every open Borel set is open. Let $\{B_i\}$ be a denumerable family of open-closed sets in a $\sigma$-space $X$. Then $\bigcup_i ...
3
votes
1answer
177 views

What is to Stone space of the free sigma-algebra on countably many generators?

I asked the question on MSE. https://math.stackexchange.com/questions/2898377/what-is-the-stone-space-of-the-free-sigma-algebra-on-countably-many-generators The answer I got, however, seems disputed....
5
votes
2answers
217 views

Complete atomless Boolean algebras with abelian automorphism group

Is there any example of a complete atomless Boolean algebra with a non-trivial abelian automorphism group? This is equivalent, by Stone duality, to asking for an extremally disconnected compact ...
0
votes
1answer
74 views

Differences between the reduced Borel field and the category algebra of a space

Let $X$ be a topological space. Halmos calls "reduced Borel field" the quotient $B(X)/M(X)$ where $B(X)$ is the Borel field of $X$ and $M(X)$ is the $\sigma$-ideal of meagre subsets of $X$. Fremlin ...
1
vote
1answer
135 views

To what logic does the free Boolean sigma-algebra of countably many generators correspond to?

The free Boolean algebra on countably many generators is closely related to the classical (two-valued) propositional calculus (after identification of logically equivalent formulas). By the Stone ...
9
votes
1answer
244 views

Embeddings of Boolean algebras in $\wp(\omega)/Fin$

If we assume MA+¬CH, then every boolean algebra with cardinality smaller than the continuum embeds in ℘(ω)/Fin. A proof of this result can be found in Theorem 1.1, Chapter 8 of the book "Hausdorff ...
1
vote
0answers
33 views

Algorithm of constructing the graph for a partial order set [closed]

Given a finite partial order set $(P,\leq)$. Is there an algorithm for constructing its graph, where, from bottom to top, the ordering goes up? Namely, I want to construct the directed graph ...
10
votes
1answer
623 views

Is every complete Boolean algebra isomorphic to the quotient of a powerset algebra?

Is every complete Boolean algebra isomorphic to a quotient, as a Boolean algebra, of some powerset algebra $\wp(X)$? It is not true for arbitrary Boolean algebras, see the comments, or see my MathSE ...
5
votes
2answers
161 views

Von Neumann's theorem on realizing automorphisms of the measure algebra

I'm looking for a proof, in English, of the following theorem due to von Neumann (which apparently originates in the paper Einige Sätze über Messbare Abbildungen, Ann. of Math, 1932): Every ...
6
votes
1answer
369 views

Boolean ultrapower of V[G] by G

In Joel David Hamkins's "Well-founded Boolean Ultrapowers as Large Cardinal Embeddings", it is mentioned that if $U \in \mathbf{V}$ is an ultrafilter of a complete Boolean algebra $\mathbb{B}$ and $U$ ...
2
votes
1answer
164 views

How to recover $k$ lost items in binary data $x_1,x_2,x_3 \dots,x_n$ via only XOR operator?

I asked this question in math.stackexchange (link) and I have had an answer for general case by using Reed-Solomon Code. More information for Reed-Solomon Coding for Fault-Tolerance in RAID-like ...
6
votes
1answer
257 views

A Bitwise Xor Problem

Consider a sequence $a_i$ defined by $$ \begin{align*} a_1&=p,\\ a_2&=q,\\ a_i&=a_{i-1} \oplus a_{i-2}+1, \end{align*}$$ where $\oplus$ is the bitwise xor operation. How can we give an ...
2
votes
1answer
91 views

Boolean completion of a partially ordered set

Given a poset $(P, \leq)$, is there a complete Boolean lattice $B$ and an order-preserving map $i_P: P\to B$ such that for any complete Boolean lattice $B'$ and order-preserving map $f: P\to B'$ ...
11
votes
3answers
554 views

Is it possible to completely embed complete Heyting Algebras into upsets of a poset?

Let $H$ be a Heyting algebra. It is a well-known result that there is a partially ordered set (Kripke frame) X such that there is an embedding of Heyting algebras $f: H \to \mathsf{Up}(X)$, where $\...
8
votes
0answers
202 views

Direct limits of $\sigma$-centered forcing notions

It is quite well known that Any FS (finite support) iteration of length $<\mathfrak{c}^+$ of $\sigma$-centered posets is $\sigma$-centered (see e.g. here). Now consider the following question: ...
6
votes
2answers
207 views

Does $\aleph_0$-density of regular open algebra entail existence of countable basis?

Suppose that the family $\mathrm{RO}(X)$ of regular open subsets of $(X,\mathscr{O})$ is a basis of $X$. Let the density of $\mathrm{RO}(X)$ (considered as boolean algebra) be $\aleph_0$. Does $X$ ...
3
votes
1answer
231 views

Stone topological Boolean algebras

I am looking for an initial reference for a theorem which is known, namely: Theorem: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological ...
1
vote
0answers
78 views

Paper by Diestel, Faires and Huff

I have been looking for a (long) while for the following paper: J. Diestel, B. Faires, and R. Huff, Convergence and boundedness of measures on non-sigma complete algebras, preprint, 1976. This ...
2
votes
0answers
48 views

Efficiently embedding finite Boolean algebras into lattices of set partitions?

Let $P_n$ be the lattice of set partitions of $[n] = \{1,2,\dots,n\}$, let $B_n$ be the Boolean algebra of subsets of $[n]$. Is there some $n_0$ such that for all $n \ge n_0$ it is possible to ...
11
votes
1answer
247 views

partitions of Boolean algebras

A partition of a Boolean algebra is a collection of pairwise disjoint nonzero elements with supremum 1. For any infinite Boolean algebra $A$ let $a(A)$ be the least size of an infinite partition of $A$...
13
votes
2answers
381 views

A curiosity on complete homomorphisms of boolean algebras

The question may be trivial, but has eluded me, may be it is more appropriate for mathstack-exchange. Let $B$, $C$ be boolean algebras and $i:B\to C$ be an homomorphism. By Stone duality to each such ...
0
votes
1answer
267 views

Representation of free Boolean sigma-algebras

By a theorem of Loomis and Sikorski, for every Boolean $\sigma$-algebra $\mathfrak{A}$ there exists a $\sigma$-field of sets $\mathcal{F}$ and a $\sigma$-ideal $\Delta$ such that $\mathfrak{A}$ is ...
2
votes
1answer
109 views

Atomicity of blocks in a Hilbert lattice

Where can I find the proof that any block (maximal boolean subalgebra) $\mathbf{B}$ of the orthomodular lattice $\mathcal{L}$ of closed subspaces of a separable Hilbert space $\mathcal{H}$ is atomic?
1
vote
1answer
264 views

Characterization of monotone boolean functions with minimum number of extremal points

Let $B = \{ 0, 1 \}$. For two points $\textbf{x}, \textbf{y} \in B^n$ we will write $\textbf{x} \preceq \textbf{y}$ iff $\textbf{x}_i \leq \textbf{y}_i$ for every $i \in \{ 1, \ldots, n \}$. A ...
2
votes
2answers
179 views

What's “serialization” really called, and is there any theory surrounding it?

Define an operator $\mathop{\vec{\bigcup}}$ as follows: Definition. Whenever $A$ is an $I$-indexed family of sets, where $I$ is a totally-ordered set, we have $$\mathop{\vec{\bigcup}}_{i \in I} A_i ...
1
vote
0answers
79 views

Weak* convergence of measures on Boolean algebras

The Dieudonné-Grothendieck theorem asserts that given a compact Hausdorff space $K$ and a uniformly bounded family $\mathcal{K}\subset C(K)^*$ (the dual to the Banach space of continuous real-valued ...
2
votes
0answers
135 views

Does $\lambda$-completion of $\kappa$-Boolean algebras preserve monomorphisms?

Let $\kappa \le \lambda$ be infinite regular cardinals. Does the free $\lambda$-completion functor $F_\kappa^\lambda$ from the category of $\kappa$-complete Boolean algebras to the category of $\...
2
votes
1answer
158 views

Logic Alphabet for more than Two Variables

Is it possible to generalise Zellweger’s logic alphabet for more than two Boolean variables? Can it be done by only using the 16 binary connectives? Thanks.
5
votes
1answer
224 views

Weak equivalence over forcing notions

We know there are several definitions about forcing equivalence which imply that two forcings notions can be equivalent or not. In general we like to know the similarity between generic extensions by ...
3
votes
0answers
121 views

A boolean representation of the Möbius function on a finite lattice

Let $(L,\wedge , \vee)$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$. Consider the Möbius function $\mu$ on $L$ defined inductively by $$\mu(\hat{1}) = 1 \text{ and } \mu(a) = - \...
4
votes
0answers
44 views

Polynomial approximations on the Boolean hypercube

Given $\vec{a} \in \mathbb{R}^n$ and $b \in \mathbb{R}$ consider the function $f(x) = Th[\vec{a}.\vec{x}+b]$ on $x \in \{-1,1\}^n$ such that the ``threshold function (Th)" gives $1$ when the argument ...
2
votes
1answer
164 views

Symmetric group acting on the set of boolean functions

Let $S_n$ act on the set of boolean functions of size $n$ in the following way: If $f$ is a boolean function and $\alpha \in S_n$, then $g=\alpha f$ and $g(x)=f(\alpha(x))$ where $x$ is boolean ...
5
votes
2answers
309 views

How “strong” is the existence of a non trivial ultrafilter on $\omega$?

Obviously the question in the title alone doesn't make sense so I'll develop on the context and then I'll ask my question : Studying $AD$ (axiom of determinacy) I had to prove that $AD$ and $AC$ are ...
4
votes
0answers
215 views

What is the algebraic equivalent of independent elements?

The definition/notion of independence is always a bit odd in measure theoretic probability theory. Definition Given a probability space $(\Omega,\mathcal{F},P)$, two sets $A,B\in\mathcal{F}$ are ...
8
votes
2answers
732 views

In what sense is GCD an extension of boolean OR?

The J Programming langauge has an operator which acts as both the GCD and boolean Or. The J Primer has this note about it: The GCD is a useful extension of the domain of the or function to non-...
2
votes
0answers
41 views

Distance measures between Boolean algebra homomorphisms

Is there a natural way to define the 'distance' between two Boolean algebra homomorphisms $f, g: B \rightarrow B'$? I'm thinking of something like the Kullback leibeler divergence for probability ...
2
votes
0answers
59 views

A question related to Boolean functions? [closed]

Let $Z_2=\{0,1\}$, $Z_2^r=Z_2\times Z_2 \times...\times Z_2,$ $S_1\subset Z_2^{n_1}$ and $S_2\subset Z_2^{n_2}$, $S=(S_2,S_1)\subset Z_2^{n_1+n_2}$, I'd like to construct $S_1$ and $S_2$ s.t the ...
1
vote
1answer
124 views

Interval topology on complete Boolean algebras

Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\...
1
vote
1answer
91 views

Missing an elementary step about the frame of strings in Barr-Diaconescu theorem

I'm trying to understand the proof of the Barr-Diaconescu theorem about Boolean covers for Grothendieck sites. Precisely, the versions you can find in Jardine's book "Local Homotopy Theory" or in Mac ...
1
vote
3answers
241 views

Is every c.c.c. non-atomic partial order of size $\omega_1$ a union of countable complete suborders?

We say that $\mathbb{P}$ is a complete suborder of $\mathbb{Q}$, if it is a suborder, and maximal antichains in $\mathbb{P}$ remain maximal antichains in $\mathbb{Q}$ As the title says, is every c.c....
4
votes
0answers
98 views

Level sets of function of inner products of vectors on hypercube

Let $H = \{ 0, 1\}^d$ be the $d$-th Cartesian product of $\{0, 1\}$ in $\mathbb{R}^d$. Suppose $v_1, \ldots, v_k$ are $k$ vectors in $H$ in general position. We define function $F \colon H^{k}\...
1
vote
1answer
90 views

Finding a set of disjoint affine subspaces such that their union is equal to a given subset of $\mathbb{F}_2^n$

Suppose I'm given a set of point $S = \{x_1, \dots, x_m \} \subseteq \mathbb{F}_2^n$, and the following task. Find a set of disjoint affine subspaces of $\mathbb{F}_2^n$, $A_1, \dots, A_k$ satisfying ...
3
votes
3answers
486 views

Incomplete subsets of the free boolean algebra on countably many generators

I know that it is provable that the free boolean algebra on countably many generators is incomplete. For the sake of concreteness, let's call the generators $p_1, p_2, p_3,...$ and refer to them as "...