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Questions tagged [boolean-algebra]

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Boolean algebras are structures which behave similar to a power set with complement, intersection and union. Use this tag for questions about Boolean algebras as structures, or about functions defined from/to Boolean algebras. For Boolean logic use the tag propositional-calculus.

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Score of 2
1 answer
223 views

I’m trying to simplify a Boolean arithmetic equation. It’s got $9$ variables, $8$ of which are symmetric. It's the Boolean logic behind the state of the "next generation" of a cell in Conway'...
Score of 2
0 answers
263 views

I am trying to "organise" Boolean expressions. I wrote a script that takes all Boolean expressions (just conjunction and disjunction without negation) for a given number of variables (here ...
Score of 0
1 answer
135 views

Let $$X:=\left\{\frac 1 n:n\in\mathbb N^+\right\}\cup \left\{1-\frac 1 n:n\in\mathbb N^+\right\}\subset \mathbb R$$ and $$Y:=[0,1)\cap \mathbb Q\subset \mathbb R$$ be the topological spaces with the ...
Score of 0
0 answers
43 views

Definition 1: A rng (pronounced "rung"), with signature $\langle S, \oplus, \underset{\oplus}{e}, \otimes\rangle$, comprises an ("additive") abelian group and a ("...
Score of 2
2 answers
111 views

The lattice $B$ of all subsets of ${\bf N}$ is a graded poset, where the rank of an element is its cardinality (as a subset of ${\bf N}$). If an element in this poset has rank $d$, then obviously its ...
Score of 2
0 answers
62 views

Lemma 22.9 of Jech says if $\kappa$ is measurable, $(B,m)$ is a measure algebra and $G$ is $B$-generic over $V$, then in $V[G]$ there is a nontrivial $\kappa$-additive measure on $\kappa$. In ...
Score of 2
0 answers
77 views

Let $(B, \mathbf{C})$ be a closure algebra and $\mathbf{I}$ the interior operator, the dual of closure operator $\mathbf{C}$. We call the set $H = \{\mathbf{I}a : a \in B\}$ with the induced algebra $(...
Score of 12
1 answer
573 views

In Boolean algebra, we have two levels of representation: (1) By Lindenbaum-Tarski representation, A Boolean algebra is isomorphic to a power set algebra iff it is complete and atomic. (2) By Stone’s ...
Score of 1
0 answers
84 views

Jech Set Theory theorem 7.13 : Let $B$ be a boolean algebra. A regular cut is a set $C \subseteq B$ that is downwards closed , and whenever $p \in B , p \notin C$ , there is a $q \leq p$ such that $...
Score of 1
0 answers
99 views

I’ve been trying to understand whether there is a common structural reason behind different “universality” results across domains. In particular, I noticed a strong similarity between: functional ...
Score of 5
1 answer
213 views

Oh, has it been 7 years? How are you mathematicians? I came back to get guidance on using Lean 4. It is really hot and crossing engineering domains. Maybe I was sleeping on it, but now I see it ...
Score of 3
1 answer
129 views

Prove $(\neg P \land \neg Q)\lor (Q \land R)\equiv (\neg P \land \neg Q)\lor (\neg P \land R)\lor ( Q \land R)$ only using logic equivalence rules? So far, I have tried to verify it with truth tables ...
Score of 4
1 answer
93 views

Let $R$ be a (not necessarily commutative) von Neumann regular ring. Let $B := \{ b \in Z(R) : b^2=b \,\}$ be the Boolean algebra of central idempotents, and write $\mathrm{At}(B)$ for the set of ...
Score of 0
0 answers
42 views

What are good references for "standard" facts in Boolean algebras, such as $x \leq (-y)+z$ if and only if $x \cdot y \leq z$, $x \cdot y = 0$ if and only if $x \cdot (-y) = x$ or similar to ...
Score of 1
0 answers
45 views

I am trying to come up with an optimisation algorithm for a set of fuzzy Boolean functions expressed in disjunctive normal form. Here is the setup. a) $n$ cases/observations $x_{1, \ldots, n}$; b) $K$ ...

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