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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Boolean rings are associative $\mathbb{Z}/2\mathbb{Z}$-algebras

Let $B$ be a boolean ring, that is, for all $x\in B$ we have $x^2 = x$. Then we know a few things about them. First $B$ has characteristic $2$, since taking $x\in B$ we have $$x+x = (x+x)^2 = x^2 + x^...
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27 views

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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59 views

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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26 views

How can I define a module over a Boolean ring?

I want to give an example of a module over a Boolean ring $R$ (in particular, the power set of a set $X$ under symmetric difference and set intersection). I thought of defining a map $(Y, m) \...
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71 views

Help understanding why finite Boolean rngs must be rings

I'm working through the exercises in Introduction to Boolean Algebra by Halmos and Givant. Looking to show the following, an exercise from the first chapter: every finite Boolean rng must have a unit. ...
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71 views

If $R$ is a Boolean ring which is in fact a field, then it must be isomorphic to $\mathbb Z / 2 \mathbb Z$ [duplicate]

Suppose $R$ is a Boolean ring ($\forall r \in R, r^2 =r$) which is in fact a field. Show it then must be isomorphic to $\mathbb Z / 2 \mathbb Z$. I already managed to show that every Boolean ring is ...
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65 views

Spectrum of a ring in which $a^p=a$ for all $a$ and prime number $p$

It is known the result that spectrum of a boolean ring is zerodimensional and compact topological space, e.g. has a base with clopen sets and satisfies Haussdorf condition. It is asked if this result ...
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57 views

Let R be a ring in which $x^2=x$ for all x ε R. (a) Prove that $x + x = 0$ for for all x ε R. (b) Prove that R is commutative. [duplicate]

I am aware this question has been asked many times but I'm still struggling to understand a few things. Let R be a ring in which $x^2=x$ for all x ε R. (a) Prove that $x + x = 0$ for for all x ε R. ...
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29 views

Irreducibility for boolean ring?

Every element in $x \in A$ in a commutative bolean ring $A$ is by definition equal to $xx$, and so therefore there are no irreducible elements in the usual sense. So I ask, how might one define ...
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48 views

How to test whether an element is in a principal ideal of a ring without enumerating all elements of the ring?

Suppose we have a finite, boolean ring $A$ induced by a finite, commutative, boolean monoid $X$ containg $0$ as in: this question. You only need the first few paragraphs of that long post. Suppose I ...
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39 views

What is an elegant way of defining complement of an element in this ring?

About the ring: $R$ is a commutative boolean ring with $1$ in which for each $x \in R$, $x + x = 0$. There is a partial order on elements of $R$ written $x \leqslant y \iff xy = y$. The ordering ...
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52 views

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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153 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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53 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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72 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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67 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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73 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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290 views

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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150 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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55 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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492 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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47 views

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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102 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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548 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
79 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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131 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
38 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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196 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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88 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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246 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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213 views

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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121 views

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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103 views

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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104 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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167 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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410 views

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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241 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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164 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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746 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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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 all $A,B,C\in P(M)...
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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 ...