Category:Congruence Relations
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This category contains results about Congruence Relations.
Let $\struct {S, \circ}$ be an algebraic structure.
Let $\RR$ be an equivalence relation on $S$.
Then $\RR$ is a congruence relation for $\circ$ if and only if:
- $\forall x_1, x_2, y_1, y_2 \in S: \paren {x_1 \mathrel \RR x_2} \land \paren {y_1 \mathrel \RR y_2} \implies \paren {x_1 \circ y_1} \mathrel \RR \paren {x_2 \circ y_2}$
Subcategories
This category has the following 5 subcategories, out of 5 total.
Pages in category "Congruence Relations"
The following 24 pages are in this category, out of 24 total.
B
C
- Condition for Equivalence Relation for Max Operation on Natural Numbers to be Congruence
- Condition for Existence of Epimorphism from Quotient Structure to Epimorphic Image
- Congruence Relation and Ideal are Equivalent
- Congruence Relation induces Normal Subgroup
- Congruence Relation on Group induces Normal Subgroup
- Congruence Relation on Ring induces Ideal
- Congruence Relation on Ring induces Ring
- Congruences on Rational Numbers
- Construction of Inverse Completion/Congruence Relation
E
- Equivalence Induced by Epimorphism is Congruence Relation
- Equivalence Relation induced by Congruence Relation on Quotient Structure is Congruence
- Equivalence Relation induced by Congruence Relation on Quotient Structure is Congruence/Corollary
- Equivalence Relation inducing Closed Quotient Set of Magma is Congruence Relation
- Equivalence Relation is Congruence iff Compatible with Operation
- External Direct Product of Congruence Relations