# Tagged Questions

Boolean algebras are structures which behave similar to a power set with complement, intersection and union. Questions regarding Boolean algebras as structures, or regarding functions defined from/to Boolean algebras fit into this tag very nicely. For Boolean logic use the tag propositional logic

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### Is this *really* a categorical approach to *integration*?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
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### Universal binary operation and finite fields (ring)

Take Boolean Algebra for instance, the underlying finite field/ring $0, 1, \{AND, OR\}$ is equivalent to $0, 1, \{NAND\}$ or $0, 1, \{ NOR \}$ where NAND and NOR are considered as universal gates. ...
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### Any two points in a Stone space can be disconnected by clopen sets

Let $B$ be a Stone space (compact, Hausdorff, and totally disconnected). Then I am basically certain (because of Stone's representation theorem) that if $a, b \in B$ are two distinct points in $B$, ...
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### Why are Boolean Algebras called “Algebras”?

Boolean algebras aren't algebras (to the best of my understanding). So why are they called algebras? Wouldn't it make more sense to call them a "Boolean system" or a "Boology" or something else like ...
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### Is XOR a combination of AND and NOT operators?

I'm not sure whether this is the best place to ask this, but is the XOR binary operator a combination of AND+NOT operators?
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### Stone's Representation Theorem and The Compactness Theorem

If you're working on $\mathsf {ZF}$ and you assume the compactness theorem for propositional logic, then you have the prime ideal theorem, and thus you can show that the dual of the category of ...
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### Duality principle in boolean algebra

All the definitions I came across so far stated, that if a statement is true, then also its dual statement is true and this dual statement is obtained by changing + ...
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### Can this set of rules perform all Boolean operations?

I never worked in this field before, I just thought about this set of rules and never saw something similar before. I apologise if I don't use the right mathematical vocabulary for my question. ...
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### Axioms for atomless Boolean algebras

I'm embarrassed to be asking, but: "Write down a set of axioms for the theory of atomless Boolean algebras." This is Exercise 1.14 in Chapter 9 of "Models and Ultraproducts" by Bell and Slomson. I'm ...
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### Definitionally equivalence between Boolean algebras and Boolean rings

On page 17, Introduction to Boolean Algebras,Steven Givant,Paul Halmos(2000): Motivated by this set-theoretic example, we can introduce into every Boolean algebra $A$ operations of addition and ...
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### Simple functions and axiom of choice

The question I have is more of a curiosity, and that is why I decided to post here instead of Mathoverflow. Before posing the question, let me set up some background. Background: Let $\Omega$ be a ...
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### Non-isomorphic countable Boolean algebras

I'm trying to solve the next exercise: Construct a sequence $\mathcal{B}_0,\mathcal{B}_1, \ldots$ of countable Boolean algebras such that for all $m \neq n$ then $\mathcal{B}_m \ncong \mathcal{B}_n$. ...
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### A free boolean algebra

Consider the following definition: The boolean algebra $A$ is generated freely with the subset $G \subseteq A$ if for every boolean algebra $B$ and map $f:G \mapsto B$ there is precisely one ...
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### Construct an OR gate when missing input information

Is there a way to construct an OR gate when the input for one combination is unknown? For example, suppose that the gate, $X$, outputs for the following inputs, $x_1$ and $x_2$, $x_1 = T$, $x_2 = T$:...
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### Is there an $O(n^2)$ test to determine if an $n \times n$ Boolean matrix $B$ has an inverse?

D.E. Rutherford shows that if a Boolean matrix $B$ has an inverse, then $B^{-1}= B^T$, or $BB^T=B^TB=I$. I have two related questions: The only invertible Boolean matrices I can find are ...
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### Can Boolean ring without unit be embedded into a boolean ring?

While going through a book (Lectures on Boolean algebra, Halmos) I got struck at the following question : Prove that every Boolean ring without a unit can be embedded in a Boolean ring with a unit. ...