The boolean-algebras tag has no usage guidance.

**3**

votes

**0**answers

88 views

### The number of monotone Boolean functions

In the paper "The number of monotone Boolean functions" A. D. Korshunov calculates an asymptotic number of the number of monotone Boolean functions
(see https://en.wikipedia.org/wiki/Dedekind_number)...

**1**

vote

**1**answer

145 views

### Finitely additive measures on Boolean algebras of regular open subsets: Is there a relationship with Borel measures? A theory of integration?

Let $\mathcal{X}$ be a topological space. An open subset $\mathcal{R}\subseteq\mathcal{X}$ is regular if it is the interior of its own closure. The intersection of two regular open sets is regular. ...

**5**

votes

**2**answers

283 views

### About a construction of Borel $\sigma$-algebra associated to a lattice

Let $(\mathcal{A}, \cup, \cap)$ a lattice (with minimum and maximum elements $\bot$ and $\top$).
Let $X\subset \mathcal{A}$ a generator set (a set of minimal cardinality that generate $\mathcal{A}$ i....

**2**

votes

**0**answers

71 views

### Effective “almost enumeration” of monotone boolean functions

Denote by $\mathcal{M}(n)$ the set of all monotone functions $\{0,1\}^n \to \{0,1\}$.
Can $\mathcal{M}(n)$ be represented as $\mathcal{M}(n) = \{ f(t) | t\in \{0,1\}^k \}$ such that:
1) $k = \log |\...

**2**

votes

**1**answer

166 views

### Boolean algebras and free filters generated by chains

Suppose $\mathbf{B}$ is a complete Boolean algebra with an infinite domain $B$. Suppose $\mathbf{B}$ is atomic (i.e. every element is the supremum of some set of atoms). This algebra contains the co-...

**1**

vote

**0**answers

47 views

### Dual equivalence for multioperators

This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...

**1**

vote

**0**answers

49 views

### Influence of independent variables on boolean functions?

Suppose a simple connected graph $G$ where its vertices are assumed to be independent. An event with uncertainty corresponds to each vertex. My instructor guides me that even though the vertices (...

**12**

votes

**0**answers

246 views

### The dominating number $\mathfrak{d}$ and convergent sequences

All spaces considered below are compact Hausdorff.
If $K$ is a space, then $w(K)$ is its weight. For a Boolean algebra $\mathcal{A}$, $K_\mathcal{A}$ denotes its Stone space. I am interested in ...

**0**

votes

**0**answers

35 views

### How to convert an expression to conjunctive normal form for Maximum-2-satisfiability?

I have a simplified Boolean expression almost ripe for maximum-2-satisfiability:
$(A\lor \neg B)\land(A\land C)\land(D\lor \neg A)$
In other words, I want to find the assignment of variables so that ...

**3**

votes

**0**answers

113 views

### Find the number of boolean functions of n variable that satisfy the following condition

For how many boolean functions is this true? The length of the shortest disjunctive normal form of that functions is equal to 2^(n-1). And the the number of variable entries in the minimal dnf of that ...

**0**

votes

**1**answer

45 views

### About equalizer of Boolean algebras

Let $A,B$ be complete Boolean algebras and $\varphi,\psi:A\rightarrow B$ be
maps preserving $0,1$, and arbitrary joins and meets. Let $C$ be the equalizer of these two; so $C=\left\{a\in A:\varphi(a)=...

**2**

votes

**1**answer

63 views

### Regular open Boolean algebras and homomorphism which does not preserve nearness of sets

I am looking for an example of topological spaces $\langle X_1,\mathscr{O}_1\rangle$ and $\langle X_2,\mathscr{O}_2\rangle$ such that
there is a homomorphism $h\colon\mathrm{r}\mathscr{O}_1\...

**8**

votes

**1**answer

182 views

### Boolean-Valued Models: Why is $\| x=y \| \cdot \| \phi(x) \| \leq \| \phi(y) \|$?

Let $B$ be a complete Boolean algebra. Jech defines a Boolean-valued model $\mathfrak{A}$ of the language of set theory to consist of a Boolean universe $A$ and functions of two variables with values ...

**2**

votes

**1**answer

87 views

### Complexity of Deciding Feasibility of a system of linear inequalities over restricted variables

I am working out an interesting problem and would like some help with this particular sub problem:
Suppose we have a matrix $ M =\left\lbrace a_{ij}\right\rbrace $ of size $n\times m$ where $ a_{ij}\...

**8**

votes

**4**answers

2k views

### Examples for “nice” Boolean algebras that are not complete or not atomic

A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms).
Boolean Algebras that are complete as ...

**5**

votes

**1**answer

95 views

### If every contraction on a Boolean algebra has a maximum value, is that Boolean algebra complete?

If $B$ is a Boolean algebra, then a mapping $f:B\rightarrow B$ is said to be contractive (or a contraction) if $f(a)+f(b)\leq a+b$ for each $a,b\in B$ where $a+b=(a\wedge b')\vee(a'\wedge b)$ is the ...

**6**

votes

**0**answers

164 views

### rigidity of $\mathcal P(\omega_1) / NS$ under MA

In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it ...

**5**

votes

**1**answer

139 views

### (A kind of) Irreducibiliy of regular open convex sets in the Cartesian space

I am looking for a proof of the fact which is formulated at the bottom of this post. The property of regular convex sets which the fact expresses seems to be true to me, yet I have not been able to ...

**4**

votes

**1**answer

187 views

### Cohen algebra and $\mathcal P(\omega)/ \mathrm{fin}$

Let $C$ be the Cohen algebra, the boolean completion of the partial order of finite partial functions from $\omega$ to 2, ordered by reverse inclusion. Does there exist an ideal $I$ on $C$ such that $...

**7**

votes

**1**answer

200 views

### Introducing meets while preserving directed closure

A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound.
Question: Suppose $\mathbb{P}$ is a separative partial order which is $\...

**6**

votes

**1**answer

168 views

### completions of regular suborders

Suppose $\mathbb{P}$ is a regular suborder of the separative partial order $\mathbb{Q}$ (see below for definitions). Must there always exist some complete boolean algebra $\mathbb{B}$ such that:
$\...

**2**

votes

**2**answers

412 views

### System of boolean equations, Satisfiability

Are there any methods to "solve" large systems of boolean equations?
$$x_{i1}\vee x_{i2}\vee x_{i3} = b_i, \quad\text{for}\quad i=1,\dots,N,$$
where $x_i, b_i \in\{0, 1\}$
For example
$$x_{1}\vee x_{...

**6**

votes

**0**answers

419 views

### Order theory as a foundation of mathematics?

I know the followings kinds of formalization of mathematics:
based on set theory (e.g. ZFC)
based on type theory (e.g. the formalism of Coq proof assistant, as an advanced example)
based on category ...

**7**

votes

**2**answers

776 views

### How do I apply the Boolean Prime Ideal Theorem?

I have become aware of an amazing phenomenon from a myriad of questions and answers here on MathOverflow: many of the results that I would typically prove using the Axiom of Choice can actually be ...

**3**

votes

**1**answer

116 views

### Are epimorphisms (defined via an obvious action) of free Boolean algebras whose set of generators is a group automorphisms?

Let $G$ be a group. Consider $B$, the free Boolean algebra with generating set (I'll call them "variables") $G$. Let $F$ be some formula (that is, some fixed element of $B$). Define an endomorphism ...

**3**

votes

**0**answers

191 views

### What algebraic identities does the iteration of forcing operation satisfy?

Let $G$ be the set of all formulas $\phi(x)$ in the language of such that $ZFC\vdash\exists x\phi(x)$ exists, $ZFC\vdash\phi(x)\rightarrow``x\,\textrm{is a complete Boolean algebra}"$, $ZFC\vdash``\...

**6**

votes

**3**answers

145 views

### Product of binary Boolean operators

I asked this question a day ago on math.stackoverflow but figured it could have an interest here.
I'm interested in the set $\mathcal{P}_N$ of boolean functions of boolean variables $p_1, p_2, \ldots,...

**6**

votes

**1**answer

188 views

### Comparing cardinalities of the spectrum of two masas in $B(H)$

If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...

**4**

votes

**2**answers

344 views

### Proving results about complete Boolean algebras in ZFC using Boolean valued models

I want to know what non-trivial ZFC theorems (not consistency results) about complete Boolean algebras (or more generally of partially ordered sets) one can prove using forcing.
I am mainly ...

**5**

votes

**3**answers

496 views

### Cohen algebra (generalization)

Let Bor($X$) = class of all borel subsets of $X$. Cohen algebra is defined as Bor(X) modulo the ideal of meager sets.
The Cohen algebra has a combinatorial : it is the unique atomless complete ...

**2**

votes

**0**answers

119 views

### Number of degree $k$ functions [closed]

Given a Boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$, there is a real multivariate multilinear polynomial that is associated with in through interpolation.
Example: $AND(x_1,x_2,\dots,x_{n-1},x_n)...

**0**

votes

**1**answer

432 views

### How to recognize if a lattice is distributive? [closed]

I know that a Boolean lattice must be distributive.
But what with these lattices? Are these distributive?
$\hskip0.7in$
How to recognize which lattices are distributive or not only by looking on ...

**1**

vote

**0**answers

44 views

### Embedding a collection of finite subsets efficiently

Are there any general non-trivial methods for solving the following problem?
Suppose one has a collection of subsets $\mathcal{C} \subseteq \mathcal{P}\mathcal{P}\{1,\dots,n\}$. They may be viewed ...

**1**

vote

**5**answers

652 views

### Is there such a thing as the sigma-completion of a Boolean algebra?

Hi all,
Suppose that $\mathcal{B}$ is a Boolean algebra. It there a way to extend $\mathcal{B}$ to a smallest Boolean algebra $\mathcal{B}'$ that contains an isomorphic copy of $\mathcal{B}$ and is ...

**1**

vote

**0**answers

189 views

### Wolfram's axiom completeness

I have been reading Wolfram's A New Kind of Science, and as I was reading the section on systems of logic and axioms, I came across this axiom, for which all of the normal axioms of Boolean logic can ...

**11**

votes

**4**answers

1k views

### Jonsson Boolean algebras?

Let us say that a mathematical structure of cardinality $\omega_1$ is Jonsson whenever every one of its proper substructures is countable.
There are examples of Jonsson groups due to Shelah or ...

**3**

votes

**1**answer

290 views

### Systems of equations in Boolean Algebra

I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by ...

**8**

votes

**2**answers

391 views

### centeredness in forcing iterations

Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$, and $B$ is $\kappa$-centered. Let $G \subset A$ be a generic ultrafilter. Is $B/G$ $\kappa$-centered in $V[G]$?
Naively, we ...

**8**

votes

**1**answer

311 views

### On $V$-decisive and weakly homogeneous forcings

Suppose that $\Bbb P$ is a forcing in $V$, we say that $\Bbb P$ is $V$-decisive if whenever $\varphi(x_1,\ldots,x_n)$ is a statement in the language of forcing, and $u_1,\ldots,u_n\in V$ then $1_{\Bbb ...

**3**

votes

**1**answer

128 views

### Boolean completion (of a forcing notion) isomorphic to each of its cones

Suppose $ \mathbb{P} := (P, {\leq_P}, 1_P) $ is a separative partial order. Let $ \mathbb{B} := \operatorname{RO}(\mathbb{P}) $ denote the Boolean completion.
Fix some dense embedding $ i \colon P \...

**-4**

votes

**1**answer

145 views

### How to find matrix representations of a boolean algebra? [closed]

Given a boolean algebra with a finite number of elements {a, b, c, ...}, and the usual operations: $\cup, \cap, \neg$.
How to find matrix representations of the elements such that:
boolean $\cup$ ...

**2**

votes

**4**answers

657 views

### Products of Boolean algebras and probability measures thereon

These are really two questions, but the second presupposes the first.
First, let $( B_i )_{i\in I}$ be an arbitrary family of Boolean algebras. I want to directly form a product of them that is like ...

**1**

vote

**2**answers

254 views

### Question on separability of a measure

Following this question here this question come to mind.
Consider a measured σ-algebra $(S,\mu)$ . Assume that μ is normalized to have total weight 1, and that S is complete (contains all subsets of ...

**1**

vote

**1**answer

103 views

### Krull dimension of dense extensions

Let $A$ be a boolean algebra and let $B\leq A$ be a boolean sub-algebra which is dense (for all $0\neq a\in A$, there is a $0\neq b\in B$ such that $b\leq a$). We suppose also that $B$, as a partially ...

**2**

votes

**1**answer

192 views

### Sigma-complete Lindenbaum algebras? [closed]

Is there any calculus whose algebraization is a sigma-complete Lindenbaum algebra, i.e., a sigma-complete Boolean algebra after identification of equivalent formulas?

**8**

votes

**3**answers

824 views

### Which Sigma-Ideals in a Sigma-Algebra are Ideals of Null Sets?

My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space ...

**4**

votes

**2**answers

125 views

### extending $\sigma$-complete boolean homomorphism

I'm not sure if this is research level, so feel free to vote to migrate.
Suppose we have a complete boolean algebra $A$, with a dense, $\sigma$-complete subalgebra $B$, and a $\sigma$-complete ...

**3**

votes

**1**answer

97 views

### existence of more distributive Boolean lattices

Are there a Boolean lattice $(X,\le)$ and a nonempty collection $(a_{ij})_{i\in I,j\in J}\subseteq X$ such that
$$\bigvee_{i\in I}\bigwedge_{j\in J}{a_{ij}}\ne \bigwedge_{f:I\to J}\bigvee_{i\in I}a_{...

**2**

votes

**1**answer

243 views

### The Universal Algebra of a sigma-Algebra

I am searching for the 'dual' algebraic structure of a Sigma Algebra. The notion of duallity is like on the case of the Boolean Algebra and Set Algera.
If X is a set, the complement and intersection ...

**-1**

votes

**1**answer

168 views

### Algebra generated by a tree [Edit] [closed]

Suppose that $(T,\leq)$ is a partially ordered set, we say $T$ is a tree* if for every $i\in T$, $\{s: s\in T, s\leq t\}$ is a well-founded chain.
What I need to know is: Can the algebra ...