# Tagged Questions

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### 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-...
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### 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 ...
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### (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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 ...
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### Algebra generated by a tree  [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 ...
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### Eigenvalues of a matrix constructed with simple logic

If a matrix can be constructed with simple bit-logic operations, is it also possible to find Eigenvalues with logic? First I'll just say that my knowledge of logic is pretty much limited to ...
A probability is a finitely additive measure on a boolean algebra with total measure $1$. A function $P:\scr B \times (\scr B - \{ 0 \})$ is a full conditional probability on $\scr B$ (for a boolean ...