# 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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### Understanding Kelley's intersection number (Boolean algebras)

It is known that: Theorem (Kelley, 1959). There exists a finite, strictly positive, finitely additive measure on a Boolean algebra $A$ if and only if $A^+$ is the union of a countable number of ...
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### Existence of a quasi-open (a.k.a semi-open) map into a Cantor cube

Recall that a topological space is extremally disconnected if the closure of any open set is open. A continuous map is quasi-open if it maps nonempty open sets onto sets with nonempty interior. For ...
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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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### 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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### 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; ...