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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Analytical origins of the Stone duality

I've asked this question in the HSM community, but by the nature of my question, some user told me to ask this question here. This is the original post https://hsm.stackexchange.com/q/13087/14296 ...
IJM98's user avatar
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An internal notion of freeness for complete Boolean algebras

Background and Definition Gaifman and Hales showed that there are no infinite free complete Boolean algebras. But let a complete Boolean algebra $B$ be internally free if there is a set $X\subseteq B$ ...
Peter Fritz's user avatar
8 votes
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146 views

On the number of Reed–Muller codewords with no consecutive ones

$\DeclareMathOperator\RM{RM}\DeclareMathOperator\Eval{Eval}$Consider the polynomial ring $\mathbb{F}_2[x_1,x_2,\dotsc,x_m]$ and let $f\in \mathbb{F}_2[x_1,x_2,\dotsc,x_m]$. Let us now fix a Gray ...
Arvind Rameshwar's user avatar
1 vote
0 answers
124 views

Minimizing all aspects of the definition of Boolean algebra

There are many equivalent ways to describe Boolean algebras. There are a number of different ways to "minimize" the description. We can: Minimize the number of function symbols. Minimize ...
Pace Nielsen's user avatar
6 votes
0 answers
151 views

Rigid boolean inclusions?

A boolean algebra $B$ is rigid if it has no nontrivial automorphisms and atomless if it has no minimal nonzero elements. $A \subseteq B$ is a complete boolean inclusion if $B$ is complete and $A$ is a ...
Doug McLellan's user avatar
6 votes
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101 views

Computing the zeta transform of a Boolean function: Space-time tradeoff

Let $f : \mathbb{F}_2^n \to \mathbb{F}_2$ be a Boolean function in $n$ variables. The zeta transform of $f$ is the Boolean function $\zeta_f : \mathbb{F}_2^n \to \mathbb{F}_2$ defined by $$\zeta_f(y) :...
Oslow's user avatar
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6 votes
1 answer
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Boolean valued models in a general setting

It is well known that Boolean valued models play significant roles for set-theoretic purposes. But how well-studied are Boolean valued models in a more general setting, as models for random first-...
Severine Climacus's user avatar
7 votes
1 answer
379 views

A set theoretic question arising from trying to understand a sheaf cohomology question

I'm trying to understand the footnote to Example 5.3 in Wiegand - Sheaf cohomology of locally compact totally disconnected spaces which is about constructing a locally compact Hausdorff and totally ...
Benjamin Steinberg's user avatar
8 votes
1 answer
420 views

Intuition behind Boolean-valued models of set theory

$\DeclareMathOperator\Card{Card}$The book Forcing Eine Einführung in die Mathematik der Unabhängigkeitsbeweise by Hoffmann provides an intuition behind boolean valued models of set theory which I will ...
Bytegear's user avatar
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Is the Pierce spectrum useful elsewhere in Mathematics?

In Borceaux and Janelidze's Galois Theories, a construction of the Pierce spectrum is given. It is the poset of ideals in a Boolean ring. It's construction is reminiscent of the Zariski spectrum in ...
Mozibur Ullah's user avatar
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Complete Boolean algebras of subsets of $\mathbb N$

Let $\mathfrak A$ be a subset of $\mathrm{Pow}(\mathbb N)$, the powerset of $\mathbb N$. Assume that $\mathfrak A$ is a complete Boolean algebra in the induced order, i.e., the inclusion order. Does ...
Andre Kornell's user avatar
7 votes
1 answer
218 views

Functions on Stone spaces as "enveloping algebra" of Boolean algebra

I'm looking for references for the following closely related facts: Given a Boolean algebra $B$, I denote by $\mathbb{Z}[B]$ the free ring generated by symbols $e_b$ such that $e_b e_{b'} = e_{b \cap ...
Simon Henry's user avatar
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What is the name for Boolean algebra's version of $\models$ between sets of identities and identities?

On p62 in Schaum's Outline of Theory and Problems of Boolean Algebra and Switching Circuits by Elliott Mendelson (1970), Part (b) of the corollary says that if an identity is satisfied by some ...
Tim's user avatar
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Boolean ring of unitary divisors / Structure of unitary divisors?

I hope this question is appropriate for MO: Let $n$ be a natural number, $U_n := \{ d | d \text{ divides } n, \gcd(d,n/d)=1\}$ be the set of unitary divisors. We can make $U_n$ to a boolean ring: $$a \...
user avatar
6 votes
1 answer
299 views

Can this ultrafilter convergence condition be expressed as a compactness condition?

Suppose that $X$ is a topological space. Let us say that an ultrafilter $\mathcal U$ on the Boolean algebra $C_X$ of clopen subsets of $X$ is partition-prime if whenever $X = \amalg_{i \in I} X_i$ is ...
Richard Garner's user avatar
2 votes
0 answers
95 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 ...
Bijco's user avatar
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43 votes
1 answer
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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 ...
Terry Tao's user avatar
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2 votes
2 answers
261 views

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(\...
ABIM's user avatar
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2 votes
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Decreasing sequences in a finitely generated closure algebra

I am interested in finitely generated closure algebras (as a special case of Heyting algebras), and in decreasing sequences of elements within such an algebra that have no lower bound. Call two ...
Andrew Apps's user avatar
16 votes
1 answer
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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 ...
Tim Campion's user avatar
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6 votes
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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?
Jiří Rosický's user avatar
10 votes
1 answer
343 views

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 $\hat{\Phi}$ ...
YCor's user avatar
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14 votes
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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\...
YCor's user avatar
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3 votes
1 answer
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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 ...
Mare's user avatar
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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{...
Rafał Gruszczyński's user avatar
5 votes
1 answer
190 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)...
Tomasz Kania's user avatar
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7 votes
2 answers
236 views

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 $\...
Monroe Eskew's user avatar
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1 answer
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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 $\...
user109711's user avatar
6 votes
0 answers
167 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\...
Helmut Bez's user avatar
6 votes
1 answer
224 views

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 ...
Toby Meadows's user avatar
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4 votes
1 answer
364 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]_{...
Ronald J. Zallman's user avatar
4 votes
1 answer
264 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 ...
LJGC's user avatar
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3 votes
0 answers
130 views

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 ...
Gro-Tsen's user avatar
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2 votes
2 answers
910 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 ...
Jon Doyle's user avatar
6 votes
1 answer
253 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 ...
Damian Sobota's user avatar
2 votes
1 answer
188 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, ...
Doug McLellan's user avatar
8 votes
4 answers
684 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?
Dominic van der Zypen's user avatar
2 votes
1 answer
171 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)?
LJGC's user avatar
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2 votes
0 answers
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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; ...
LJGC's user avatar
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1 vote
0 answers
111 views

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 $\...
Arrow's user avatar
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3 votes
1 answer
116 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.) ...
Michal Walicki's user avatar
2 votes
0 answers
144 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 ...
Zoorado's user avatar
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1 vote
1 answer
155 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 ...
Miroslav Korbelar's user avatar
2 votes
2 answers
276 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 \...
Zoorado's user avatar
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8 votes
5 answers
463 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,...
Tomasz Kania's user avatar
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1 vote
2 answers
216 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 ...
Dominic van der Zypen's user avatar
1 vote
1 answer
613 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 ...
Beginner's user avatar
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5 votes
1 answer
399 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....
Beginner's user avatar
  • 175
6 votes
2 answers
466 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 ...
Iian Smythe's user avatar
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0 votes
1 answer
136 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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