2
votes
1answer
41 views

How to solve this boolean algebra problem?

Given two expressions: $$A\bar{D}+A\bar{C}D +A\bar{B}C + ABCD = Y$$ and $$BD+A\bar{C}D=Z$$ is there a way to simplify this using the rules for Boolean Algebra? I tried different combinations, but if I ...
1
vote
1answer
28 views

Can covering be done on two elements?

The covering rule is: $$B \bullet (B+C) = B$$ and $$B+(B \bullet C)=B$$ So does it follow from this rule that: $$B \bullet A \bullet \bar{C} + B \bullet D \bullet\bar{F} = B \bullet ...
1
vote
1answer
24 views

Arrow's Impossibility Theorem Using Boolean Algebra

I am currently working on a research project which involves using Boolean matrices for the proof of Arrow's Impossibility Theorem and various other lemmas and results related to quasi ordered sets. In ...
1
vote
1answer
24 views

Boolean-like algebra

Suppose one had an algebra that that follows most of the laws of Boolean algebra (associative, commutative, distributive, identity, annihilator, idempotent, double negation, De Morgan) but does not ...
2
votes
1answer
61 views

What does quotienting by a congruence mean?

I have come across quotient algebras in my different mathematics courses. I know of quotienting with normal groups, quotienting with ideals etc. While studying Boolean Algebra I encounter quotienting ...
1
vote
1answer
53 views

Sigma Algebra: Etymology

Why do we talk of sigma algebras in measure theory. As far as I know sigma is related to the countability. But what does it stand for?
2
votes
1answer
17 views

Boolean matrices and Algebra

Let us consider, a set of binary rectangular matrices of finite dimensions, call the set as $T$. The cardinality of the set $T$ is $2^{mn}$ where each matrix are of order m cross n. Suppose $S$ is a ...
0
votes
0answers
33 views

How to show that two (Boolean) algebras are NOT isomorphic?

Suppose that we have two algebras $A$ and $B$. To show such algebras are isomorphic, all we have to do is constructing a bijective homomorphism between them. Not for negative case, we should check all ...
2
votes
2answers
65 views

Can Boolean ring without unit be embedded into a boolean ring?

While going through a book (Lectures on Boolean algebra, Halmos) I got struck at the following question : Prove that every Boolean ring without a unit can be embedded in a Boolean ring with a unit. ...
0
votes
1answer
30 views

Boolean algebra (ISO)

Let $\mathfrak{A}$ be a Boolean algebra and $E$ be an element in $\mathfrak{A}$. The set of all subelements of $E$ forms a Boolean algebra, denoted by $\mathfrak{A}_E$. Suppose that $I$ be the ...
0
votes
1answer
21 views

Explain one statement about Stone Space

In this page Stone Space This is no clear for me "The points in S(B) are the ultrafilters on B, or equivalently the homomorphisms from B to the two-element Boolean algebra" I think I means every ...
2
votes
1answer
78 views

Commutative ring addition where $a^2 = a$

I'm trying to solve following question: If $a^2=a$ for all $a \in R$ where $R$ is a commutative ring, then $a+a=0$. I have tried to solve this problem for a while now and I'm more or less stuck. I ...
2
votes
1answer
40 views

Looking for an algebraic structure

I'm looking for the name of algebraic structures (in which the elements are partially ordered) with the following properties: Top element defined, bottom optional; Join defined for all elements, ...
0
votes
1answer
65 views

Condition for the boolean algebra of clopen sets to be extremely disconnected.

Let $X$ be a topological space and let $\Gamma \mathcal O(X)$ be it's boolean algebra of clopen subsets. For compact totally disconnected space, show that $\Gamma \mathcal O(X)$ is complete (as a ...
1
vote
1answer
54 views

Where can I learn more about these two functions obtained from IFF and XOR?

Given a set $X$, write $\mathrm{heaps}(X)$ for the set of all finite heaps (or 'multisets', if you prefer) on $X$. Under this definition, it is well-known that if a binary operation $*$ on a set $X$ ...
4
votes
1answer
77 views

Ideal:Kernel :: Filter:?

I understand that the notion of a filter is in some sense dual to the notion of an ideal, at least in the context of Boolean algebras1. Let $f:{\mathbf A} \to {\mathbf B}$ be a Boolean algebra ...
2
votes
1answer
140 views

Can boolean homomorphisms of boolean algebras correspond to ultrafilters?

I am trying to solve 5th problem in Exercises 2.9 in Awodey's book on page 55: Show that for any boolean algebra $B$, boolean homomorphisms $h : B \to 2$ correspond exactly to ultrafilters in $B$. I ...
1
vote
1answer
55 views

Is it true “Every Boolean algebra is an algebra of sets, for any given set X”

I have confused between these two notions, please help Every Boolean algebra is an algebra of sets, for any given set $X$, and the converse is false.
1
vote
0answers
109 views

Showing that a Boolean algebra is a Boolean ring

I've proved that a Boolean ring is a Boolean algebra but I am having trouble with the converse. The operation for + is defined as the symmetric difference for elements $a$ and $b$ from the Boolean ...
1
vote
2answers
68 views

Partially Ordered Sets question

For $m\in\mathbb{N}$, which integers are covered by $m$? I've been playing with the prime factors of $m$ and I can't seem to see any pattern. Can anyone help?
5
votes
1answer
95 views

Adjoint for functor involving Boolean rings

Let $R$ be a a commutative ring with a unit element, then one can associate to $R$ a Boolean ring $B(R)$, as in this text by Bergman, last line of page 594. (I guess this is a very classical thing. ...
1
vote
1answer
147 views

Terminology question; inverse vs complement in Boolean algebra

This was said at a lecture I attended: $e$ is neutral element for operation $*$ if $\forall x (x*e=x \wedge e*x = x)$. So, for example 0 is n. e. for disjunction and 1 is n. e. for ...
2
votes
1answer
75 views

Which of the following representations of Karnaugh map is 'better'?

I usually come across two representations of Karnaugh maps in books and on the web as shown in the figure. The difference is whether the higher order variables are on the rows or on the columns. I ...
1
vote
1answer
94 views

Localizations of the Boolean Ring P(X)

Given a set $X$, we can construct the Boolean ring whose elements are the power set of $X$. The multiplication therein is intersection, and the addition is symmetric difference. I am interested in ...
2
votes
1answer
159 views

What does Tarski mean by a “tautological operation” on a Boolean algebra?

I am reading Part II of Chin and Tarski's "Distributive and Modular Laws in the Arithmetic of Relation Algebras". In the beginning of section 4, the authors say "In general, if $\odot$ is a binary ...
3
votes
4answers
1k views

An “atom” in Boolean algebra

Could someone explain what an atom in Boolean algebra means? I am acquainted with ring theory and group theory but not Boolean algebra. As far as I can tell from browsing around, it is something like ...
0
votes
2answers
1k views

What do these terms mean: commutative, associative, distributive

I am reading a book, and I am trying to understand what the writer really mean by the following terms. I would like to understand what these words mean in relation to the examples. In regular ...
1
vote
1answer
1k views

Parity Checking and truth tables

I have a question that I am very confused about. Parity Checking. Produce a truth table for a parity checking circuit that is based on $4$ input data bits, an input parity bit and a single ...
6
votes
1answer
258 views

Simple functions and axiom of choice

The question I have is more of a curiosity, and that is why I decided to post here instead of Mathoverflow. Before posing the question, let me set up some background. Background: Let $\Omega$ be a ...
3
votes
1answer
202 views

What's a structure where $a-a = 1$ and $a\cdot a^{-1} = 0$?

I'm trying to specify a structure that has the basic features of a Boolean algebra but not necessarily restricted to binary sets. However, I observed that I almost have a field except that ...
0
votes
3answers
275 views

Equality in Boolean Algebra

Say, $$A = C \lor (C\land D) = C \land(1\lor D) = C$$ $$A = C \lor (C\land D) = (C\lor D)\land(C\lor C) = C\land(C\lor D)$$ Now, the part I don't understand here is if we equate we get: $$C \land ...
28
votes
1answer
1k views

Universal binary operation and finite fields (ring)

Take Boolean Algebra for instance, the underlying finite field/ring $0, 1, \{AND, OR\}$ is equivalent to $ 0, 1, \{NAND\} $ or $ 0, 1, \{ NOR \}$ where NAND and NOR are considered as universal gates. ...
1
vote
2answers
168 views

Quotient ring and Boolean algebra correspondence

While reading a probability paper titled Gröbner bases and factorisation in discrete probability and Bayes (can't find free version, sorry), I came across the explanation: The set of indicator ...