Questions tagged [congruence-relations]
For questions about general congruence relations, i.e. equivalence relations on an algebraic structure that are compatible with the structure. Please DO NOT use this for questions about integer modular arithmetic.
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Congruence relation in monoids
Suppose $(M, \ast)$ is a monoid and $\sim$ is a congruence relation on $M$, i.e. $\sim$ is an equivalence relation such that: $x_1 \sim y_1$ and $x_2 \sim y_2$ imply that $x_1 \ast x_2 \sim y_1 \ast ...
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Congruences on the pentagon lattice $\mathcal{N}_5$
Let $\mathcal{N}_5$ refer to the Pentagon lattice, or the lattice generated by the set $\{0, a, b, c, 1\}$ subject to $1 > a$, $1 > c$, $a > b$, $b > 0$ and $c > 0$.
My aim is to find ...
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If every congruent class contains only one element, then the set is finite.
I would like to ask a question to clarify a query I posted yesterday on every numerical Semigroup is finitely generated. Let's start with the following:
Suppose we have a set $ A \subseteq \mathbb{N} $...
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Congruence lattice of a semiring
A famous result of Funayama and Nakayama states that the congruence lattice of any lattice is a distributive lattice [1]. Also, it can be proved that the lattice is a frame/ complete Heyting algebra 2....
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Identifying group of units of monoid to a point
Let $S$ be a monoid. Let $\mathfrak{g}(S)$ be the group of units of $S$, and denote by $S/A$ the quotient $S/\sim_A$ where $\sim_A$ is the smallest congruence such that $a\sim_A a'$ for all $a, a'\in ...
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Defining Addition/Multiplication on $\Bbb Z_3$
On $\Bbb Z_3$, we typically define addition and multiplication as follows:
$$[a]+[b]=[a+b]$$ $$ [a]\cdot[b]=[a\cdot b]$$
Consider defining addition as $[a]+[b]=[0]$ for all $a,b\in \Bbb Z$. This ...
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Is there alternative symbol for congruence classes?
I saw on youtube a talk about the congruence classes. For example, the congruence classes modul0 4 are written as followings.
$[0]_4 = \{0, 4, 8, 12, \dots \} $
$[1]_4 = \{1, 5, 9, 13, \dots \} $
$[2]...
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Exercise 4, Section 1.5 of Hungerford’s Abstract Algebra
Let $\backsim$ be an equivalence relation on a group $G$ and $N=\{a\in G\mid a\backsim e\}$. Then $\backsim$ is a congruence relation on $G$ if and only if $N$ is a normal subgroup of $G$ and $\...
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Exercise 3, Section 1.5 of Hungerford’s Abstract Algebra
Definition: Let $(G,\circ)$ be a group. A congruence relation on $G$ is an equivalence relation $\equiv$ on the elements of $G$ satisfying if $g_1\equiv g_2$ and $h_1\equiv h_2$, then $g_1\circ h_1\...
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When does this construction always yield a congruence?
Suppose $\mathcal{M}$ is a commutative monoid and $E$ is any equivalence relation on $\mathcal{M}$. Define $\widehat{E}$ to be the "shift-invariant" part of $E$, that is, $$a\widehat{E}b\...
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Linear congruential generator and hyperplanes
Given $x_{n+1}=1229x_n \pmod{2048}$ and $u_n=x_n/2048$, I have to find the number of lines on which the points $(u_n,u_{n+1})$ lie.
I know that by Marsaglia's theorem, this will be $(2048\cdot 2)^{1/2}...
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Combinatorial proof of binary partition function $b(n)$ is always even
For all integer $n$, let $b(n)$ be the number of partition of $n$ into power of two.
For instance, $b(4)=4$, since
\begin{align*}
4 &= 2^2 \\
&= 2^1+2^1 \\
&= 2^1+2^0+2^0 \\
&= 2^0+2^0+...
user1089451
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Congruence properties of binary partition function
For every positive integer $n$, denote $b(n)$ be the number of binary partition of $n$, i.e., the number of partition of $n$ into power of two, where the power is decreasing.
For instance, $b(5)=4$ ...
user1089451
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Principal congruence of a distributive lattice.
The following problem is taken from Burris and Sankappanavar's A Course in Universal Algebra (11, pg 42).
Suppose $L$ is a distributive lattice and $a,b,c,d\in L$.
Then $\langle a, b\rangle\in\Theta(c,...
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Every equivalence relation $x \equiv y$ compatible with the structure of a module $E$ is of the form $y-x \in M$ for some submodule $M$ of $E$.
On the end of page 196 of Bourbaki’s Algebra I, it says:
Let $E$ be an $A$-module.
Every equivalence relation $x \equiv y$ compatible with the structure of a module $E$ is of the form $y - x \in M$ ...
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