Questions tagged [boolean-algebras]

A Boolean algebra is a commutative ring satisfying x²=x for every x, and sometimes required to have a unit; they have characteristic 2. For coding theory (notably dealing with subsets linear subspaces of spaces of Boolean functions), rather use the [coding-theory] or [linear-algebra] tag.

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73 views

Projective objects for compact po-spaces

Let us consider the following definition: a compact po-space is a pair $(X,\leq)$ where $X$ is a compact space and $\leq$ is an order, closed on $X^2$. Then, we can consider the category $KPoSp$ whose ...
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Adjoints to the inclusion of complete Boolean algebras in complete Heyting algebras

Let $\bf{Bool}$ be the full subcategory of $\bf{Hey}$, the category of complete Heyting algebras and meet,joins and implication preserving maps between them. Is $\bf{Bool}$ a (co)reflective ...
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1answer
1k views

In the category of sigma algebras, are all epimorphisms surjective?

Consider the category of abstract $\sigma$-algebras ${\mathcal B} = (0, 1, \vee, \wedge, \bigvee_{n=1}^\infty, \bigwedge_{n=1}^\infty, \overline{\cdot})$ (Boolean algebras in which all countable joins ...
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Existence of measurable “inclusion” into Euclidean space

Let $(\Omega,\mathfrak{F})$ be a measurable space. When does there exist an injective measurable function $f:(\Omega,\mathfrak{F})\to (\mathbb{R}^n,B(\mathbb{R}^n))$ to some Euclidean space, here $B(\...
14
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1answer
709 views

What is a module over a Boolean ring?

Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...
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221 views

Reduced products of complete Boolean algebras

I expect that complete Boolean algebras are not closed under reduced products modulo $\kappa$-complete filters, for any regular cardinal $\kappa$. Is it true? And, is a reference for this?
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Elementary equivalence between $n\mapsto n+1$ and its inverse on the Stone-Čech remainder?

Consider structures $(A,f)$ encoding a Boolean algebra $A$ endowed with an automorphism $f$. There is an obvious notion of isomorphism between such structures. Consider the endomorphism $\hat{\Phi}$ ...
11
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1answer
272 views

Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?

Let $\beta\omega$ be the Stone-Čech compactification of the discrete infinite countable space $\omega$, and $\beta^*\omega=\beta\omega\smallsetminus \omega$ is the Stone-Čech remainder. The map $j:n\...
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1answer
120 views

Reference for Dedekind's problem

Dedekind's problem is about enumerating antichains in the Boolean lattice. Is there an explicit reference where Dedekind stated this problem? Is there a good motivation to study this problem except ...
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1answer
154 views

A problem of non-emptiness of intersections of certain chains of regular open sets

Let $X$ be a topological space and $\mathrm{RO}(X)$ its complete boolean algebra of regular opens. Define well inside relation: $$U\prec V\iff\overline{U}\subseteq V.$$ Let $\mathcal C\subseteq\mathrm{...
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1answer
167 views

Do 1-additive maps admit tensor products?

Let $\mathcal{F}$ be a set algebra (or a Boolean algebra). Following Kalton, let me call a function $f\colon \mathcal{F}\to \mathbb R$ $\delta$-additive ($\delta \geqslant 0$), whenever $f(\varnothing)...
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Linear suborders of $(P(\omega),\subseteq)$

Consider the partial order $(P(\omega),\subseteq)$. Let $L$ be a dense linear suborder. Does $L$ have a countable dense subset? (Note that it contains a copy of $\mathbb R$, via Dedekind cuts of $\...
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1answer
144 views

Ordering a subset of the clopens of a Stone space

Let $P$ be a countably infinite set of propositional variables and $\mathcal{L}_P$ be the propositional language generated from $P$ and the usual connectives $\wedge$, $\neg$, $\vee$. The set $\...
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153 views

Boolean functional equations

My current approach to investigating reversible quantum gates requires the solution of Boolean functional equations. For example, $$f(x,y,z) = f(x,y \oplus f(x,y,z), z \oplus f(x, y, z)),$$ where $f\...
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1answer
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Is the category of atomless Boolean algebras with complete embeddings closed under coproducts?

Consider the category whose objects are atomless Boolean algebras (not necessarily complete) and whose arrows are complete embeddings. Does a coproduct exist in this category for any two atomless ...
4
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1answer
256 views

Functor from rings into compact Hausdorff spaces

There is an adjunction $\text{Bool}^{op} \leftrightarrow \text{Set}$ between boolean algebras and sets which sends a boolean algebra to the set of its prime ideals and a set $X$ to $[X, \mathbb{F}_2]_{...
4
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1answer
166 views

Infinite distributive laws in atomless free sigma-algebra

Let $\frak{A}$ be the free $\sigma$-algebra on $\omega_1$ free $\sigma$-generators. Then $\frak{A}$ is not completely distributive because it is atomless. However, is it $\omega$-distributive in the ...
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Open sets on a Stone space

If $B$ is a Boolean algebra (possibly assumed complete), is there a standard name for the Heyting algebra (or frame) $L := \Omega(S(B))$ of open sets on the Stone space $S(B)$ of $B$, — or for the ...
2
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2answers
180 views

Is the intersection of Boolean sublattices a Boolean sublattice?

Let $L$ be a boolean lattice, $A$ and $B$ sublattices of $L$ that are themselves boolean lattices, and suppose that $I = A \cap B$ is nonempty. Is $I$ a boolean sublattice of $L$? Is it a ...
6
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232 views

Finite covers of Boolean algebras by their subalgebras

It is a student exercise that no group can be represented as a set-theoretic union of its two proper subgroups. The same also can be shown for Boolean algebras. On the other hand, it's not hard to ...
2
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1answer
144 views

Semi-rigid boolean algebras

A boolean algebra is rigid if it has no nontrivial automorphisms. Call it semi-rigid if none of its nontrivial automorphisms has any fixed points other than 0 and 1.* The four-element algebra $\{0, b, ...
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4answers
482 views

Are there $2^{\aleph_0}$ pairwise non-isomorphic Boolean algebra structures on $\omega$?

Is there a collection of $2^{\aleph_0}$ pairwise non-isomorphic countable Boolean algebras? Equivalently, are there $2^{\aleph_0}$ pairwise non-homeomorphic closed subsets in the Cantor space?
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1answer
138 views

Separability of the Stone space of a free sigma-algebra

Let $X$ be the Stone space of the free $\sigma$-algebra $A$ on $\omega_1$ free generators. Is $X$ separable (i.e. does $X$ contain a countable dense set)?
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3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators

During my studies, I came across several different Stone spaces, e.g.: (i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators; ...
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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 $\...
3
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1answer
103 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
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135 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 ...
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1answer
141 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 ...
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2answers
261 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 \...
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5answers
437 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,...
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2answers
123 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 ...
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1answer
302 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 ...
5
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1answer
273 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
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2answers
297 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 ...
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1answer
126 views

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

Let $X$ be a topological space. Halmos calls "reduced Borel algebra" 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 $X$. ...
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1answer
204 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 ...
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1answer
285 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 ...
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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 ...
11
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1answer
939 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 ...
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2answers
183 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
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1answer
421 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
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1answer
213 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 ...
7
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1answer
293 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
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1answer
136 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 $f: P\to B'$ ...
12
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3answers
613 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 $\...
9
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289 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
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2answers
229 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
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1answer
269 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 ...
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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 ...
2
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1answer
87 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 ...