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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.

For congruence relations on groups, rings, vector spaces, universal algebras etc.

A congruence relation is a relation on an algebraic structure, which is compatible with given operations. Congruences of an algebra form the congruence lattice.

Congruence relations on groups correspond to normal subgroups, congruence relations on rings correspond to ideals.

255 questions
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$\mathbf{N}_5$ as a congruence lattice

A finite lattice is said to be representable if there exists a finite algebra whose congruence lattice is isomorphic to that lattice. As I was reading a paper, I came across the line: "The reader can find examples to show that every lattice with…
Eran
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$f(x) \equiv 1$ mod $(x-1)$ and $f(x) \equiv 0$ mod $(x-3)$ then is there any $f(x)$?

Let $S$ be the set of polynomials $f(x)$ with integer coefficients satisfying $f(x) \equiv 1$ mod $(x-1)$ $f(x) \equiv 0$ mod $(x-3)$ Which of the following statements are true? a) $S$ is empty . b) $S$ is a singleton. c)$S$ is a finite non-empty…
cmi
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"Universal" differential identities

I now asked this at MO. Let $f:\mathbb{R}^d \to \mathbb{R}$ be smooth. The mixed derivatives commute: $f_{xy}=f_{yx}$. This identity is "universal" in the sense that it holds for any smooth map. Question: Are there any universal identities which…
Asaf Shachar
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Lattices are congruence-distributive

$\newcommand{\r}[1]{\mathrel{#1}}$ First, a few definitions. Given a lattice $L$, a congruence on $L$ is an equivalence relation $\theta$, compatible with the lattice operations, i.e. if $x_1\r{\theta}x_2$ and $y_1\r{\theta}y_2$, then $x_1\wedge…
Miha Habič
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Associanize a magma

This is a thing I have been thinking on and gotten a bit frustrated so I share my thoughts here in hope for clarification. Let $M$ be a magma, that is a set with an underlying binary operation which we denote $\cdot$. The binary operation is not…
Zelos Malum
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What is the interpretation of $a \equiv b$ mod $H$ in group theory?

I.N. Herstein has defined: Let $G$ be a group, $H$ a subgroup of $G$; for $a,b \in G$ we say $a$ is congruent to $b \mod H$, written as $a \equiv b \mod H$ if $ab^{-1} \in H$. Let $G$ be a group, $H$ a subgroup of $G$; for $a,b \in G$ we say $a$…
user88923
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Should we teach congruences from the start rather than normal subgroups in group theory?

Many students have difficulty initially with the idea of a quotient group, and why a normal subgroup is defined as it is. I think this is potentially made worse by the slightly obscure logic of the subject as presented via normal subgroups. Could…
Joshua Tilley
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Quotienting a group by an equivalence relation

In Jacobson's book, Basic Algebra 1, he points out that you can form quotient groups by looking at the equivalence classes of a 'multiplication compatible' equivalence relation on the group. By multiplication compatible, he means that if $A,B$ are…
RougeSegwayUser
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What does mod and congruence mean in algebra.

I have sometimes seen notations like $a\equiv b\pmod c$. How do we define the notation? Have I understood correctly that $c$ must be an element of some ring or does the notation work in magmas in general?
curious
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How can I know the number of solutions of the following congruence $x ^ {10} \equiv 1 \pmod {2016}$?

How can I know the number of solutions of the following congruence $x ^ {10} \equiv 1 \pmod {2016}$? My professor quick answer was: $$\mathbb{Z_{2016}^*} \cong \mathbb{Z_{32}^*} \times \mathbb{Z_{9}^*} \times \mathbb{Z_{7}^*} \cong (C_{8} \times…
Intuition
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Finding all non trivial congruence relations on (N, +)

I want to find all congruence relations on ($\mathbb{N}, +$). Clearly, we have $\bigtriangledown_\mathbb{N} = \mathbb{N} $ x $ \mathbb{N} $ and $\bigtriangleup_\mathbb{N} = \{(x,x)| x \in \mathbb{N}\}$. How do I determine non-trivial ones? How can…
Thomas Poisson
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$M_3$ is a simple lattice

I'd like to prove (exercise 9.5 in Roman's Lattices and Ordered Sets, p.203) that the lattice $M_3$ is simple, meaning that the only congruences on $M_3$ are the trivial ones (the 'equality' congruence, i.e. $\{(x,x); x\!\in\!M_3\}$, and the…
Leo
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What can we actually do with congruence relations, specifically?

Let $T$ denote a Lawvere theory and $X$ denote a $T$-algebra. Under my preferred definitions: A subalgebra of $X$ consists of a $T$-algebra $Y$ together with an injective homomorphism $Y \rightarrow X$. A quotient of $X$ consists of a $T$-algebra…
goblin GONE
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Congruence lattice of $N_5$

I calculated the the congruence lattice of $N_5$ using hit and trial and then verified it with Universal Algebra calculator. But I need to prove that it is the congruence lattice of $N_5$ How should I do that? Given below is the Con Lattice.
Alvis
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Group congruences: If the operation is preserved, do we get $a\sim b$ $\Rightarrow$ $a^{-1}\sim b^{-1}$?

Let $(G,*)$ be a group. Let $\sim$ be an equivalence relation such that $$(\forall a,a',b,b'\in G)a\sim a', b\sim b' \Rightarrow a*b\sim a'*b'. \tag{*}$$ I.e., the equivalence relation $\sim$ respects the group operation. Question. Does the…
Martin Sleziak
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