# Questions tagged [boolean-ring]

Use this tag for questions related to Boolean rings such as the ring of integers modulo $2$ $\mathbb Z/2\mathbb Z$.

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### Finite Commutative ring with 100 elements where $x^2=x$?

Does there exist a finite Commutative ring with 100 elements where $x^2=x$ for every $x\in R$? I know finite Boolean rings has the property this property but they have cardinality $2^n$, for some $n$. ...
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### Boolean Ring has $2^N$ elements "proof"

I was thinking about a proof that all finite booleans rings has $2^N$ elements. My attempt was the following: Consider a boolean ring and all of its isolated elements, i.e., elements that isn't a sum ...
94 views

### A boolean ring which is local must be isomorphic to $\Bbb{F}_2$.

Recall: a Boolean ring is a (commutative) ring $R$ where $\forall x \in R: x^2=x$. I don't really know how to proceed. I have tried some things like If $x,y \ne 0$ in $R$ such that $x^2=x$ and $y^2=y$...
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### Prove the Fourier coefficients $c_S$ of $f\colon\mathbb{F}_2^n\to\mathbb{F_2}$ is $\sum_{\operatorname{supp}(x)\subseteq S}f(x)$

Given $n>0$. Suppose $f:\mathbb{F}_2^n\to\mathbb{F_2}$ can be expressed as $f(x)=\sum_{S\subseteq [n]}c_Sx^S \pmod{2}$, where $x=(x_1,\cdots,x_n)^T, c_S\in\mathbb{F}_2,x^S=\prod_{i\in S}x_i$. It is ...
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### Has a group of exponent $2$ a structure of a boolean ring?

It is known that, for a boolean nonzero ring (i.e. all elements are idempotent), the subiacent additive group is a group of exponent $2$ (thanks to @Arturo for pointing out, a group of exponent $2$ is ...
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### Why is every Boolean ring regular?

I answered my own question. As usual, I was overthinking. In writing this up, I figured it out, so thought I'd keep it as a Q&A, seeing as though I couldn't find it anywhere. The Question: Show ...
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### Complete Subalgebras and Dense Subalgebras of Complete Boolean Algebras are Regular Subalgebras

In Thomas Jech's Set Theory book, he states that If $A$ is a complete subalgebra of a complete Boolean algebra $B$ then $A$ is a regular subalgebra of $B$. Also, if $A$ is a dense subalgebra of $B$ ...
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### Proving from De Morgan's laws

I was proving: $J.K'.L+J.K.L'+JKL=J.(L+K)$ I have done the following: From LHS: $J.K'.L+J.K.L'+JKL$ $=J.(K'.L+K.L'+K.L)$ $=J.(K'.L+K.(L'+L))$ Since $L'+L=1:$ $=J.(K'L+K)$ [' is complement] Now I'm ...
1 vote
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### A question concerning supremum in a Boolean ring

In a Boolean $\sigma$-ring, is it true that for every sequence {$x_n$} and y, we have $supx_n\cdot y = sup(x_n\cdot y)$ ? Background: I was reading 'Measure Theory' written by Paul R.Halmos (in the ...
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### Number of truth outputs of a Boolean function

Disjunctive Normal Form (DNF) A Boolean function $f$ of $n$ variables is said to be in a DNF if it is a disjunction of conjunction in $n$ variables. That is, each conjunction includes all the ...
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1 vote
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### Number of terms in a DNF equals the number of truth outputs

In a Boolean function $f$, it seems that the number of terms in a Disjunctive Normal Form (DNF) is equal to the number of truth outputs of $f$. For example, If $f=A+B$, then clearly, $f$ has $2^2-1=3$ ...
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### Given a boolean matrix $M$ what are the matrices formed by replacing $1s$ in $M$ with $0s$ called?

Given two boolean matrices $A$ and $B$ over some common dimensions one can form an order via $A\leq B\iff \forall i,j(A_{i,j}\leq B_{i,j})$ under this order, what would the matrix $A$ be called in ...
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### Why is the boolean "OR" operator denoted as "+"?

I learnt boolean algebra as part of a computer hardware course where the focus was very much on using it as a foundation for creating digital logic blocks out of gates, so there was very early on ...
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### Equivalent properties of a proper ideal of a generalized boolean algebra

I do not understand the item c) of the following question, the exercise 9 from section 1.2 from the book "Lattice-ordered Rings and Modules" from Stuart A. Steinberg: A generalized boolean algebra is ...
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### Proving associativity of a certain binary aperation in any complemented distributive lattice

If, in a Boolean lattice $(X,\vee,\wedge,0,1,')$ (i.e. a complemented distributive lattice), we define $x+y=(x'\wedge y)\vee(x\wedge y')$, is there an elegant way to prove that $(x+y)+z=x+(y+z)$ ...
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### Rings in which each element is a sum of $n$ commuting idempotents

Let $n$ be a nonnegative integer. Let $R$ be a nonunital ring such that every element of $R$ is a sum of $n$ pairwise commuting idempotents. (As usual, the class of nonunital rings includes the class ...
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