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Questions tagged [quotient-set]

This tag should be used for questions about quotient sets that might not have any other quotient structure or quotient structures that don't have their own tag.

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Nonunital non commutative ring with 3 ideals...

It is well known that if a (unital commutative) ring A has only three ideals ({0}, J, A), then the quotient A/J is a field. But, what can we conclude about A/J if A is not commutative nor unital but ...
Felipe's user avatar
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0 votes
1 answer
23 views

Does the quotient set uniquely determine arbitrary homogenous relations?

It seems totally reasonable to me to generalize the quotient of equivalence relations to general ones: Given a binary relation $R$ over sets $X, Y$, each $x \in X$ has an $R$-class, denoted $[x]_R = \...
n1lp0tence's user avatar
2 votes
1 answer
50 views

Is there a name for the arc $\mathbb{S}^1 / (x \sim360 - x)$

I was playing with some ideas in a vague way and I have encountered this structure that arises from taking the space of angles $\mathbb{S}^1$ and quotienting it by the relation $(x, 360-x)$ (here $360$...
Sidharth Ghoshal's user avatar
0 votes
1 answer
38 views

How are "types" defined in this module?

Note: This might end up being a question about a simple concept that I forgot about (I am very tired at the time of writing this, after all), so maybe try skipping to the bottom. I'm learning about ...
iwjueph94rgytbhr's user avatar
0 votes
1 answer
89 views

What is the cardinalities of the quotient set and individual abstraction classes?

The given equivalence relation is defined as $r \subseteq P(\mathbb{N})^2$: $P \ r \ Q$ if and only if $P = Q = \emptyset$ or $P, Q \neq \emptyset$ and $\min P = \min Q$. What is the cardinality of ...
user avatar
0 votes
1 answer
41 views

Quotient set of an "equivalence relation" without reflexivity

Let $\sim$ be some symmetric and transitive relation on a set $X$. Couldn't the notion of a quotient set still be defined, where some elements may "vanish"? For example, let $A=\{1,2,...,7\}$...
Samuel Han's user avatar
5 votes
4 answers
649 views

Is left multiplication well defined on a quotient set of a group?

Let $G$ be a group and $H$ a subgroup, now for every $g \in G$ we define $\sigma_{g}:G/H \rightarrow G/H: xH \mapsto gxH$. Note that we know nothing about the subgroup $H$. My question is whether or ...
H. de Gracht's user avatar
-2 votes
1 answer
65 views

Quotient Ring example issue

I am currently learning about lattices and came across a Mr.Bill Buchanan video that explains the Ring-LWE problem. I understand how modulo works and quite understand the math behind it. However, I ...
A. H's user avatar
  • 1
2 votes
1 answer
71 views

Definition and construction of a quotient space using a toy example

To make headway into Agebraic Topology I need to precisely understand the definition and construction of the quotient space. Definitions in my textbooks and online have felt handwavy to me, and I don'...
Nate's user avatar
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0 votes
2 answers
38 views

Is $X/{\sim}:=\{[x]: x\in X\}=\emptyset$ or $X/{\sim}:=\{[x]: x\in X\}=\{\emptyset\}$ when $X=\emptyset$?

Let $X:=\emptyset$. Suppose $R\subset X\times X$. Then, $R=\emptyset$. $R$ is a relation (the only relation) on $X$. $R$ is an equivalence relation. Let ${\sim} := R$. Let $[x]:=\{y\in X:y\sim x\}$ ...
tchappy ha's user avatar
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0 votes
0 answers
60 views

Relation between $\mathbb{Q}$ and $\mathbb{Z}^2$

Is there a way of writing $\mathbb{Q}$ in terms of $\mathbb{Z}^2$ in a simple expression? My idea is to construct $\mathbb{Q}$ a quotient set of $\mathbb{Z}^2$, where $\mathbb{Z}$ is the set of ...
sam wolfe's user avatar
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1 vote
1 answer
137 views

Quotient set with respect to another set

The usual definition I have seen for a quotient set of some set $S$ is with respect to some equivalence relation $\sim$, i.e. $S / \sim$, which means all elements $x$ and $y$ where $x \sim y$ are ...
Joel Croteau's user avatar
1 vote
1 answer
134 views

Theorem on the basis of a field extension $F[x]/(p(x))$ where $p(x)$ is irreducible

I am reading through Dummit and Foote's Abstract Algebra, and while I'm usually pretty good with my ring and field theory, this theorem is giving me some trouble. Let $p(x)$ $\in F[x]$ be an ...
MajorMath's user avatar
0 votes
2 answers
44 views

The Basis for the set of cosets (V/U) does not count for (u + U), where u is an element of U (and V)

I havent found any similar questions regarding my interests, feel free to correct me if needed. My confusion comes from a theorem in my math notes, which goes as follows: Let $B_1$ be a basis for U, ...
Green Ideology's user avatar
0 votes
1 answer
75 views

The function that takes two quotient sets and merges them

I want to know the definition and the well-definedness of the function that takes two quotient sets (disjoint-set data structures), merges them and returns a quotient set. For example, if the function ...
Jin SANO's user avatar
0 votes
0 answers
56 views

Quotient space given an equivalence relation

Given $\mathbb{R}^2$ with its usual topology, we define the following equivalence relations and I'm asked for the quotient spaces: a)$(x_1,y_1)R(x_2,y_2)$ iff $x_1^2+y_1^2=x_2^2+y_2^2$ I understand ...
Valere's user avatar
  • 1,344
0 votes
0 answers
32 views

Understanding why a quotient map is open [duplicate]

There seems to be something that I am fundamentally misunderstanding about quotient maps as I cannot convince myself that given a quotient map $f:X\to Y/R$ is open. To be specific, let $(X, \tau)$ be ...
Epsilon Away's user avatar
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3 votes
0 answers
82 views

Why are monadic categories over $\mathsf{Set}$ cocomplete?

$\newcommand{\set}{\mathsf{Set}}\newcommand{\T}{\mathcal{T}}$Given any monad $(\T,\eta,\mu)$ over $\set$, it is claimed that the Eilenberg-Moore category of algebras $\set^\T$ is cocomplete. More ...
FShrike's user avatar
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3 votes
0 answers
110 views

Doubt in a paper involving factor graphs

I am reading from the paper here. Let $X$ be a graph and $G$ a finite group which acts on it (as per the definition in the paper, the group action is by default faithful). Now, we say that an edge $e=\...
user avatar
2 votes
3 answers
126 views

Understanding the definition of cones in James Dugundji's.

James Dugundji defines (Topology, chap. VI definition 5.1) a cone in the following manner For any space $X$, the cone $TX$ over $X$ is the quotient space $(X\times I)/R$, where $R$ is the equivalence ...
Choripán Con Pebre's user avatar
1 vote
0 answers
83 views

Quotient set of $\mathbb R^2/SO(2)$

I am trying to understand quotient sets with an example of rotations in the plane. I have seen that $\mathbb R^2/SO(2)\cong \mathbb R_{>0}$. As I understand this, the quotient set is the set of ...
RedPen's user avatar
  • 581
-1 votes
1 answer
54 views

$R=\Bbb{R}[x,y]/(x+y-a,x^2-ax+b)$ is integral domain when $a^2-4b<0$. [closed]

Let $a,b∈\Bbb{R}$ and let $R=\Bbb{R}[x,y]/(x+y-a,x^2-ax+b)$. I want to prove that if $a^2-4b<0$ , then $R$ is integral domain. My try and thougt; This condition means $x^2-ax+b$ is irreducible, I ...
Poitou-Tate's user avatar
  • 6,351
0 votes
1 answer
35 views

Is this proof correct to show that $\text{Im}(\varphi)$ is countably infinite?

I want to show that the image of the totient function $\varphi: \mathbb{N} \to \mathbb{N}$ is countably infinite. Notice that $\varphi = i \circ b \circ p$ where $p$ is a surjective map, $b$ is a ...
Joshua Ortiz's user avatar
3 votes
1 answer
365 views

Doubt on the notation of quotient sets and quotient vector spaces

In pure set theory level, to construct the quotient set we need: $1)$ A set $X$ $2)$ A equivalence relation $\thicksim$ $3)$ With $2)$ and $3)$, we can form the pair $(X, \thicksim)$. $4)$ With $(X, \...
M.N.Raia's user avatar
  • 895
0 votes
2 answers
127 views

What group action underpins the quotient by a group action in this answer?

What group action underpins the quotient by a group action in this answer? What's the group and what's the operation? I know what a quotient is. I know what a group action is. I have looked up what ...
it's a hire car baby's user avatar
1 vote
1 answer
321 views

Concept of quotient map and quotient topology.

I am studying quotient topology.Different books define the concept differently but I think I have got the crux of the concept.I want to verify whether I have understood properly and whether it is ...
Kishalay Sarkar's user avatar
2 votes
1 answer
70 views

If $A$ has no non-trivial idempotents, then neither does $A/N$

Let $A$ be a commutative, associative, unital, finitely generated algebra over an algebraically closed field $k$. Denote by $N$ the nilradical of $A$, which is the set of all nilpotent elements of $A$ ...
MyWorld's user avatar
  • 2,438
1 vote
1 answer
197 views

Showing that pushout with inclusion induces closed map into pushout

How do i prove this formally correct? For $X,Y$ Topological Spaces and $A \subset Y$ closed. Let the Following Diagram Commute: $\require{AMScd}$ \begin{CD} A @>{i}>> Y\\ @V{f}VV @VVV\\ X @&...
bigbusiness's user avatar
3 votes
1 answer
103 views

Process of identifying quotients of polynomial rings

I'm taking an abstract algebra class and we are introducing quotient rings specifically polynomial quotient rings and I'm trying to work out some example problems, but I cannot figure out a general ...
Jacob Sanders's user avatar
4 votes
1 answer
146 views

Is it true that $\mathbb{Z}[\zeta_8]/\langle 1+3\sqrt{i}\rangle\cong\mathbb{Z}_{82}$?

I've been thinking a lot about ideals and factor rings, and I came upon the following. First, consider $\mathbb{Z}[\zeta_8]$, which is the cyclotomic ring of integers such that all $z\in\mathbb{Z}[\...
AKemats's user avatar
  • 1,337
2 votes
1 answer
111 views

What kind of structure is $\mathbb{Z}[i]/\langle 2+2i\rangle$?

Consider the factor ring $R=\mathbb{Z}[i]/\langle 2+2i\rangle$, where $\langle 2+2i\rangle$ is the ideal of the Gaussian integers such that for all $z\in\mathbb{Z}[i]$, $z(2+2i)\in\langle 2+2i\rangle$....
AKemats's user avatar
  • 1,337
0 votes
1 answer
29 views

Characterizing the product-structure on cosets $O_{h}/C_{2v}$

I consider the set of cosets $Q=G/H$ where $G=O_{h}\simeq S_{4}\times Z_2$ is the full octahedral symmetry group and $H=C_{2v}\simeq Z_2 \times Z_2$, the symmetry group of a body with a $C_2$-axis in ...
BeMuSeD's user avatar
  • 105
0 votes
0 answers
41 views

Finding the quotient set of relation

Let A={1,2,3,4,5,6,7,8,9}. We define (a,b)R(c,d) iff 2|(a-c) ∧ 3|(b-d). R is an equivalence relation on the set AXA. find the quotient set. I got 2 equivalence classes - [(1,3)], [2,4] how do I know ...
Omer Mualem's user avatar
3 votes
1 answer
78 views

Products in quotient category

Suppose $\mathcal{C}$ is a category, $\sim$ is a congruence relation on $\mathcal{C}$, and $[-]: \mathcal{C} \to \mathcal{C}/{\sim}$ is the quotient map. I'm able to prove that if $\mathbf{0}$ is ...
Jordan Barrett's user avatar
1 vote
0 answers
44 views

"Quotient" of category by arbitrary reflexive relation

I went through the definition of the quotient of a category $\mathcal{C}$ by a congruence relation $\sim$ on the morphisms. I was very surprised to find that you can prove $\mathcal{C}/{\sim}$ is a ...
Jordan Barrett's user avatar
0 votes
1 answer
43 views

Existence of a function from a quotient

On "Introduction to Smooth Manifolds" by John M. Lee, at page 309 we're talking about multilinear algebra. There's a proposition on the Characteristic Property of the Tensor Product Space. ...
Turquoise Tilt's user avatar
0 votes
2 answers
378 views

Proving there exists a unique function decomposition

I am trying to prove the following result. Let $X$ and $Y$ be sets and $\sim$ an equivalence relation on $X$. Let $f: X \to Y$ be a function which preserves $\sim$, and let $\pi$ denote the ...
Brad G.'s user avatar
  • 2,248
1 vote
1 answer
316 views

Thinking about the cosets of $\mathbb{Z}[X]/(f(x))$

Let's think about $$\mathbb{Z}[X]/(f(x))$$ First of all, (f(x)) is the ideal generated by $f(x)$ on $\mathbb{Z}[x]$. Let's consider $f(x) = x^d+1$. Then, if we ...
Guerlando OCs's user avatar
2 votes
1 answer
92 views

How to prove that a function on a quotient set $\mathbb{Z}/n\mathbb{Z}$ is well defined?

I am asked to prove that a function under a quotient set is well-defined. I know that for a function to be well-defined, two mappings or images found in the co-domain/range can't be mapped from the ...
Khalil Alashy's user avatar
1 vote
1 answer
116 views

Special case where $I\subseteq J$ implies $R/J\subseteq R/I$ seems to hold

Consider a ring $R$ and two ideals $I,J$ such that $I\subseteq J$. There is a natural projection $R/I\to R/J$ by the isomorphism theorem. There is, in general, no homomorphism going the other way ...
mrtaurho's user avatar
  • 16.2k
1 vote
1 answer
42 views

Constructing a quotient set for negative numbers

There's a relation $\rho$ defined as $\forall a,b \in \mathcal{A}$ such that $\mathcal{A} = R - \{0\} $ and $(x,y)\in \rho = x \cdot y > 0$. I proved that this is in fact an equivalent relation and ...
ranu's user avatar
  • 179
0 votes
2 answers
230 views

How to show $\frac{R}{J} \subset \frac{R}{I}$ when $I \subset J$?

I know the elements of $\frac{R}{J}$ are like $r+J$ when $r \in R$. So I have to show that there is some $s \in R$ that $r+J = s+I$. But I don't know how to do that. I actually want to show that $\...
Mina's user avatar
  • 576
0 votes
0 answers
38 views

How do I construct setoids in terms of coequalizers?

Many constructive theorem provers don't have native support for quotient types. In order for presheafs over Set to work properly I need some sort of workaround. Currently I'm using a simple approach ...
Ms. Molly Stewart-Gallus's user avatar
0 votes
1 answer
322 views

Quotient set of the relationship $x,y\ \in \mathbb{Q}, \ x \sim y \iff x-y \in \mathbb{Z}$

$x,y\ \in \mathbb{Q} \ \\ x\sim y \iff x-y \in \mathbb{Z}$ I already know this is an equivalence relationship, in fact: reflexive: $x-x =0 \in \mathbb{Z}$ symmetric: $x-y\in \mathbb{Z} \implies y-x \...
Loris Simonetti's user avatar
1 vote
1 answer
49 views

Show $\mathbb R^{\ge 0}$ can be taken as a model for $\mathbb R^2/\sim$

Paolo Aluffi says something to the effect of what's in the title in his intro to math notes. Link below. $\mathbb R^2/\sim$ is a quotient of $\mathbb R^2$ modulo the equivalence relation given by $(...
user882358's user avatar
0 votes
1 answer
105 views

A consequence of quotient map from unit circle to itself

In Part II of General Topology, Munkres, Theorem 57.1 states that if $h:S^1\to S^1$ is continuous and antipode-preserving, then $h$ is not nulhomotopic. There are some definitions I should clarify ...
kelvin hong 方's user avatar
3 votes
0 answers
125 views

How general is the fundamental theorem of equivalence relations?

We know that the fundamental theorem of equivalence relations can be stated without reference to sets by using congruence as a conjunction of two propositions as following: If $R$ together with ...
Jaspreet's user avatar
  • 759
2 votes
3 answers
189 views

Composition of equivalence relations and quotient set

Let $R$ be an equivalence relation on non empty set $X$. Prove that $X/(X/R\circ R \circ \dots R) = R$ Since $R$ is an equivalence relation we have $$R\circ R \circ \dots R = R$$So we should prove ...
S.H.W's user avatar
  • 4,359
2 votes
1 answer
75 views

Visual understanding of $\frac{\mathbb R} {\mathbb Q} $

Consider the quotient set $\frac{\mathbb R} {\mathbb Q} $ obtained by the equivalence relation $$ x \equiv y \mod \mathbb Q \quad \text{iff} \quad x-y \in \mathbb Q $$ I was wondering if there exists ...
Adriano Banchieri's user avatar
0 votes
1 answer
59 views

Let $X = [0,1]$ and $\sim$ have the following equivalence relation over $X$: is the quotient first countable?

Let $X = [0,1]$ and $\sim$ have the following equivalence relation over $X$: $$x\sim y\iff x = y\text{ or }x, y \in {]}0, 1{[}$$ Show that $X/{\sim}$ is not first countable. =========================...
Andre's user avatar
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