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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70 views

Commutativity of Boolean ring

I could not prove that "Every Boolean ring is commutative", but I found it on the Internet. I am giving the idea of the proof that I "learned". Let $R$ be the Boolean ring. We can easily prove that $...
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Construct algebras $A_p \subseteq \mathscr(P) (X)$ with cardinality $|A_p | = 2^p$.

If $X$ is a set with $|X| \geq s$, for every $p \in \{1,2,...,s\} $, construct an algebra $\mathscr{A} _p \subseteq \mathscr{P} (X)$ such that $|\mathscr{A} _p | = 2^p$. I was thinking on doing this ...
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Boolean equations simplified..

e. Create suitable diagrams to represent four derived logical operations using correct notation. These should include the truth tables and icons used for following gates.? i. NAND ii. NOR iii.XOR iv....
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41 views

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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59 views

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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62 views

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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69 views

Shall we call this relation an “isomorphism”?

This is a short question: Assume the values: $\mathbf{True}$, $\mathbf{False}$ and the logic symbols: $\land,\lor$ Is $\mathbf{True}$, $\mathbf{False}$ under $\land,\lor \,$ isomorphic to $\...
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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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69 views

GRE 9768 #60 Boolean non-commutative rings: Prove $(-s)^2=s^2$ without commutativity.

GRE 9768 #60 Ian Coley's approach is to prove $(I)$ and $(I) \implies (II) \implies (III)$ In proving $(I)$, how do we prove $$(-s)^2=s^2$$ without commutativity (but with $s=s^2$, if need ...
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GRE 9768 #60 1. Does $(s+t)^2=s^2+t^2$ imply $s+s=0$? 2. Idempotent matrices do not form a ring?

GRE 9768 #60 on what appears to be Boolean rings: Ian Coley's approach is to prove $(I)$ and $(I) \implies (II) \implies (III)$ I think $(II) \implies (I)$. My attempt: $$(s+t)^2=s^2+t^2 \...
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49 views

Set Ring and Algebraic Ring

Let $\mathscr{R}$ be a ring of sets and define set operations $\odot=\text{multiplication}$ and $\oplus=\text{addition}$ by $$E\odot F=E\cap F$$ and $$E\oplus F=E\triangle F$$ then $\mathscr{R}$ is ...
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478 views

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. ...
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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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46 views

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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268 views

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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1answer
56 views

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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93 views

A non-Boolean ring without unity with this property

I'm exploring rings that have the following property: $\rule{10cm}{0.4pt}$ For all $x \in R$, and for all $n \in \mathbb{N}$, there exists $y \in R$ such that $\sum_{i=1}^{n} x = y \cdot x$. $\rule{...
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1answer
37 views

Expressing real function algebraically for every point in its domain

Let's say that f(x) is a function with its domain called $A$ with its codomain labeled $B$. The indicator function $I_{\{p\}}(x)$ has a value of the multiplicative identity, $1$, when $x=p$, and $0$ ...
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71 views

Compliment multiplication Compliment

Since, $a+a'=1$ and $a.a'=0$ (Boolean Algebra complement laws), where $a$ is a variable and $a'$ is its complement. Can you please explain what $a'.a'=$ would be? Also, please prove your result as ...
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73 views

Boolean algebra : number of atoms, unions, intersections

Is there a formula that relates the number of atoms of a boolean algebra to the number of different unions/intersections of some generating elements? Consider the boolean sub-algebra $A$ generated by ...
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177 views

Question about Boolean rings.

As we know every finite boolean ring in which $1≠0$ is isomorphic to, $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$. So, is every infinite boolean ring is isomorphic to $\...
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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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1answer
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What algebraic structure best fits the Cantor set?

Let $2^{\Bbb N}$ be the set of infinite binary sequences $\{x_n\}$ where $x_n\in \{0,1\}$ for every $n\in \Bbb N$. I want it to fit the axioms of a known algebraic structure such that the following ...
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1answer
84 views

Is the Cantor set (binary sequences) with logical operations a ring?

Can a Boolean ring be somehow extended to infinite binary sequences? If $2^{\Bbb N}$ is the Cantor space (the set of all infinite binary sequences), is it a ring if the operations are defined term by ...
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139 views

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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163 views

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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117 views

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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499 views

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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1answer
45 views

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 ...
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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 ...