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
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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 $X\subseteq B$ ...
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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 ...
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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 ...
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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 ...
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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) :...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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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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(\...
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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 ...
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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 ...
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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?
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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 $\hat{\Phi}$ ...
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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\...
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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 ...
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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{...
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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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1
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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 $\...
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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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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 ...
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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]_{...
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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 ...
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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 ...
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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 ...
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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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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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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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0
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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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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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1
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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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2
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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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2
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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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1
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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 ...
5
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1
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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 ...
0
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1
answer
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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 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 $X$.
...