Questions tagged [boolean-ring]

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

Filter by
Sorted by
Tagged with
2 votes
2 answers
84 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 $y^2=y$...
soggycornflakes's user avatar
0 votes
1 answer
42 views

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 ...
qmww987's user avatar
  • 861
2 votes
2 answers
123 views

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 ...
Paul Rebenciuc's user avatar
2 votes
1 answer
100 views

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 ...
Shaun's user avatar
  • 44.4k
0 votes
1 answer
59 views

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 $B$ ...
Ali Dursun's user avatar
2 votes
1 answer
73 views

The Boolean lattice of a Boolean ring

I am proving that a Boolean Ring is also a Boolean Lattice. I defined $\leq$ as $x\leq y$ when $xy=x$. The supremum is $a+b+ab$, and infimum is $ab$. The Max element is $1$, Min is $0$. I proved that $...
ShishRobot's user avatar
2 votes
0 answers
86 views

Proof verification: Classical Propositional Logic is Post-Complete

Proof verification: classical propositional logic is Post-complete. I'm trying to prove the Post-completeness of classical propositional logic. In order to do this, I will be proving a theorem about ...
Greg Nisbet's user avatar
  • 11.6k
0 votes
2 answers
318 views

Is the Boolean Algebra on two elements {0,1} a ring, field, or both?

I am aware of the difference between field and ring, and I have also read other posts on this site posing similar questions on Boolean Algebra but have been unsatisfied with the focus of the questions ...
Scully Maywood's user avatar
1 vote
0 answers
83 views

Star-autonomous categories are categorifications of Boolean algebras?

1. Question The n-Lab article on the Chu-construction says: "Armed with just this much knowledge, and knowledge of how star-autonomous categories behave (as categorified versions of Boolean ...
Max Demirdilek's user avatar
2 votes
2 answers
953 views

Prime ideal $\implies$ maximal in a Boolean ring

I want to show that a prime ideal in a non-unital Boolean ring $B$ is maximal ideal. If the ring contains unity then it is easy. As Boolean rings are commutative, for a prime ideal $P$ the ring $B/P$ ...
Infinity_hunter's user avatar
0 votes
1 answer
101 views

The Boolean ring $\mathscr{P}(\Bbb N)$ has an ideal $\mathscr{P}(\Bbb N_+)$. Find the elements of $\mathscr{P}(\Bbb N)/\mathscr{P}(\Bbb N_+)$.

Denote $\Bbb N_+ = \Bbb N \setminus\{0\}$. The Boolean ring $\mathscr{P}(\Bbb N)$ has an ideal $\mathscr{P}(\Bbb N_+)$. Find the elements of $\mathscr{P}(\Bbb N)/\mathscr{P}(\Bbb N_+)$ and compute the ...
Rico Jello's user avatar
0 votes
1 answer
193 views

Are Boolean rings integral domains? [duplicate]

Are Boolean rings integral domains? My assumption was no. The "product" in Boolean rings is the intersection $\cap$ of two sets from $\mathscr{P}(X)$. If Boolean rings we're integral ...
Rico Jello's user avatar
0 votes
0 answers
148 views

Ring with every element idempotent

I am having query in this MCQ. Let $R$ be a ring such that every element is idempotent then (a) Every prime ideal is maximal ideal. (b) Every maximal ideal is prime ideal (c) if $|R|> 2$ implies $...
math student's user avatar
  • 1,171
0 votes
2 answers
59 views

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 ...
MsBonniePython's user avatar
1 vote
0 answers
39 views

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 ...
nauxuyuh's user avatar
0 votes
1 answer
91 views

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 ...
gete's user avatar
  • 1,352
1 vote
1 answer
392 views

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 $2^2-1=3$ ...
gete's user avatar
  • 1,352
3 votes
1 answer
94 views

A property of Boolean algebra

In a Boolean algebra $\mathcal B$, we know that $$x+\bar{x}y=x+y\text{ for all } x, y\in \mathcal B.$$ By following the above identity, we can also write $$xy+\bar{x}yz=xy+yz.$$ Can we write $$\bar{y}...
gete's user avatar
  • 1,352
1 vote
1 answer
62 views

commutative ring to boolean algebra

Let X be a set. We know that $(P(X),\triangle, \cap )$ is a commutative ring with the zero-element $ \emptyset $ and the one-element $X$. $P(X)$ is the power set, $\triangle$ the symmetrical ...
trsommer's user avatar
1 vote
0 answers
111 views

Define an isomorphism between finite boolean ring and $\mathbb{Z}_2\times \cdots\times \mathbb{Z}_2$

I'm trying to prove any finite boolean ring $R$ is isomorphic to $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$. I know this question is already on this site, but the proofs I saw used the ring's ...
Alejandro Bergasa Alonso's user avatar
0 votes
1 answer
135 views

Is Zero ring a Boolean ring?

I have a simple doubt. Is zero ring a Boolean ring? ( We do have $0 ^2 =0$ ). Or do we assume a Boolean ring to be non-zero?. The doubt came in my mind while showing that in a Boolean ring every prime ...
Mohit Sharma's user avatar
0 votes
1 answer
31 views

What is the algebraic structure of all the quadtrees under these operations?

I was implementing a quadtree in a programming language, and I realized I could define operations such as negation $\bar{x}$, reunion $a\cup b$ and intersection $a \cap b$ on these objects. These ...
rambi's user avatar
  • 215
2 votes
2 answers
25 views

Identify $S = 0$ for any set $S$ and form a non-boolean ring out of $\Delta$ on sets?

Let $S$ be a set and define for each $A \subset S$ the negative of $A$ to be $-A = S \setminus A$. So that $A + S\setminus A = S$. Thus we identify $0 = S$. The $+$ operation is still $\Delta$ so ...
Daniel Donnelly's user avatar
0 votes
1 answer
48 views

Show that atom of Boolean algebra B inside B' is also the atom of B' where B' is subalgebra of B. [closed]

Let B := (B, ≤, ∨, ∧, c , 0, 1) be a Boolean algebra and B' := (B' , ≤, ∨, ∧, c , 0, 1) be a Boolean subalgebra of B. Show that an element of B0 that is an atom of B must also be an atom of B' . ...
gg2121's user avatar
  • 15
0 votes
1 answer
35 views

What is the motivation behind defining $[\alpha] + [\beta] = [(\alpha \land \lnot \beta) \lor (\lnot \alpha \land \beta)]$?

Consider $\mathcal{F}$ to be the set of all well-formed formulae (wffs), and $\mathcal{F}/\equiv$ denotes the set of equivalence classes under the equivalence relation induced by $\equiv$ (logical ...
stoic-santiago's user avatar
0 votes
1 answer
205 views

Proof verification: prime ideals in Boolean rings

This question have been answered many times but I didn't find a similar proof, I submit it here for verification. Let $R$ be a Boolean ring (i.e. $\forall r\in R, \exists n>1: r^n=r$). Prove that ...
Conjecture's user avatar
  • 3,066
0 votes
1 answer
124 views

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^...
SeraPhim's user avatar
  • 1,168
1 vote
0 answers
35 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 ...
user3865391's user avatar
  • 1,345
2 votes
1 answer
84 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 ...
Rasmus Källqvist's user avatar
2 votes
1 answer
63 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) \...
Manj's user avatar
  • 229
5 votes
3 answers
182 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. ...
omegaplusone's user avatar
0 votes
1 answer
738 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 ...
user avatar
2 votes
1 answer
293 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 ...
jpatrick's user avatar
  • 914
1 vote
2 answers
621 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. ...
SFR's user avatar
  • 445
2 votes
1 answer
40 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 ...
Daniel Donnelly's user avatar
1 vote
1 answer
242 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 ...
Daniel Donnelly's user avatar
0 votes
1 answer
90 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 ...
Daniel Donnelly's user avatar
1 vote
0 answers
117 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 ...
user avatar
1 vote
3 answers
693 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 $...
MUH's user avatar
  • 1,377
1 vote
1 answer
81 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 ...
Daniel Kawai's user avatar
  • 1,005
2 votes
1 answer
99 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 $(x+y)+z=x+(y+z)$ ...
Daniel Kawai's user avatar
  • 1,005
2 votes
1 answer
106 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 ...
darij grinberg's user avatar
0 votes
1 answer
77 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 $\...
Andrew's user avatar
  • 2,031
7 votes
2 answers
544 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,...
user avatar
2 votes
1 answer
213 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 ...
BCLC's user avatar
  • 13.4k
2 votes
1 answer
252 views

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 \...
BCLC's user avatar
  • 13.4k
-1 votes
1 answer
155 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 ...
Pedro Gomes's user avatar
  • 3,841
1 vote
2 answers
542 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. ...
Fabius Wiesner's user avatar
0 votes
2 answers
79 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 ...
John tired's user avatar
0 votes
1 answer
207 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? ...
user544158's user avatar