# Questions tagged [boolean-algebras]

The boolean-algebras tag has no usage guidance.

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### Notions of connected components in a finite family fibration

Let $\Pi_0:\mathsf{FinFam}(\mathsf C)\to \mathsf{FinSet}$ be the fibration exhibiting the free finite coproduct completion of $\mathsf C$. Suppose $\mathsf C$ has finite limits so that the extensive $\...

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### How to prove de Vries algebras morphisms are dense and full if their duals are into?

Well, this is a quite short question but I think it will require some explainations.
Let's say that a de Vries (or compingent) algebra is a Boolean algebra $B=(B,0,1, \wedge, \vee, \neg) $ with a ...

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### 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.) ...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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,...

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### 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 ...

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### 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 ...

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### 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....

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $U$ ...

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### 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 ...

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### 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 ...

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### 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 $f: P\to B'$ ...

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### 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 $\...

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### 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: ...

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### 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 $X$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $A$...

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### 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 ...

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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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.

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### 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 ...

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### 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) = - \...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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-...

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### 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 ...

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### 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 ...

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### 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\...

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### 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 ...

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### 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....

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### 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}\...

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### 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 ...