# Questions tagged [boolean-ring]

Use this tag for questions related to Boolean rings such as the ring of integers modulo 2.

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### Finite Boolean ring with $1 \neq 0$ is isomorphic to $\mathbb{Z}_2\times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$ [duplicate]

Let $R$ be a finite Boolean ring with $1 \neq 0$. Show that $R\cong \mathbb{Z}_2\times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$. This is exercise 2 on p267 in the book abstract algebra of ...
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### How is it possible for a set of (algebraic normal form) monomials to have disjoint support?

We call $supp(F) = \{v \in V : F(v) = 1\}$ the support of F. Neumann, 2006, page 5. (http://www.mathematik.uni-kl.de/~dempw/Thesis/neumann.pdf) This is the standard definition of "support of a ...
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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 $(x+y)+z=x+(y+z)$ ...
1answer
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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 ...
2answers
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### An $m$-ary function that represents all $n$-ary functions

Motivation It is well-known that any binary operator $*$ on the boolean ring $\{0,1\}$ can be represented using only one of the $\operatorname{NAND}$ and $\operatorname{NOR}$ operators. For example,...
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### Boolean algebra/ring: Compute product of sum of combinations of variables

Let's say I have a set of variables $a_1, ... ,a_n \in \{0, 1\}$ and a boolean algebra where multiplication is $\land$ and sum is $\oplus$ (exclusive or). This is like in bit operations in computers. ...
2answers
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### On Boolean algebras and Atoms

I know plenety of questions have been made around this topics, but i've got an specific one. If $\mathfrak A$ Is an infinite boolean algebra, and $C$ is a finite subset of atoms in $\mathfrak A$, then ...
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### Are these boolean rings integral domains?

Let X be a set and let P(X) denote the Boolean ring whose elements are the subsets of X, with addition being symmetric diﬀerence and multiplication being intersection. Is P({1}) an integral domain? ...
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### Show that a Boolean ring is a commutative ring. [closed]

I have a question about (https://math.stackexchange.com/q/10279)'s proof to this. I also asked as a comment but I am unsure whether it will be replied to since the post was made 8 years ago? My ...
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### How to prove convergence of a sequence of binary numbers

I have a boolean expression with 4 inputs and 1 output, that when iterated onto itself(output->input(s)), the function converges to 1. How do I go about proving the convergence of a sequence of binary ...
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### Understanding boolean rings

I'm reading a book "Introduction to Abstract Algebra" by Neal McCoy. I've come across a few exercises which discuss "Boolean rings". The text defines Boolean rings as: A ring $R$ is a Boolean ...
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### A Boolean ring in which if $2a=0$ then $a=0$

In every Boolean ring we have $a^2=a$ for every $a$ in the ring. In some Boolean rings, if $2a=0$ then $a=0$. How to show this ring has just one member? Thanks in advance
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### Stone duality for sets

Let $X$ be a set. Then, $\operatorname{Mor}_{\mathbf{Set}}(X,\{ 0,1\} )$ is a Boolean ring. In fact, it is a complete atomic Boolean ring and this, together with the the cofunctor from complete ...
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### Calculating Walsh Hadamard Transform

Can anyone say me the steps involved in calculating Walsh Hadamard Transform $(W_f(a))$ for a Boolean function $f(x)=x^3$ in a finite field $GF=F_4$?
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### How to find the probability of truth of the following boolean expression?

$a_1(a_4+a_8a_5+a_8a_7a_6)+a_2(a_8a_4+a_5+a_7a_6)+a_3(a_6+a_7a_5+a_7a_8a_4)$ Take the above boolean expression, a function of $a_1$ to $a_8$, each of which are independent and is zero by probability ....
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### A finite subset of a Boolean algebra, generates a finite subalgebra.

Let $\mathscr{B}$ be a Boolean algebra and let $F$ be a finite subset of $\mathscr{B}$. Is $\left<F\right>$, the subalgebra generated by $F$, necessarily a finite subalgebra of $\mathscr{B}$? I ...
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### why is the set of intersections and symmetric differences of a fixed set finite

I have a question very similar to Finite ring of sets For a given set U, the set R of subsets is a ring under the operations of symmetric difference ($\bigtriangleup$) and intersection ($\cap$). ...
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### Power Set of $X$ is a Ring with Symmetric Difference, and Intersection

I'm studying for an abstract algebra exam and one of the review questions was this: Let $X$ be a set, and $\mathcal P(X)$ be the power set of $X$. Consider the operations $\Delta$ = symmetric ...
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### Ring of Sets vs Ring in Universal Algebra

I actually want to continue this post. I believe the naming convention in Mathematics is consistent, such that there are no clearly distinguish objects have the same name. However, I'm not sure how to ...
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### Is there a program which shows set equality in steps?

To show: $(P(M),+,\cdot)$ is a commutative ring. $A+B:= (A\cup B) \backslash (A\cap B)$ (or XOR) $A\cdot B:=A\cap B$ First Ring axiom: Addition is assiociative $(A+B)+C=A+(B+C)$ for ...