Knowra Congruence relation Congruence relation A congruence relation is an equivalence relation on an algebraic structure that is preserved by each of its operations. It allows equivalent elements to be identified while retaining well-defined operations on equivalence classes.
Equivalence relation : A binary relation that is reflexive, symmetric, and transitive. A congruence is an equivalence relation with an additional compatibility condition.
Group congruence : A congruence relation on a group, preserved by multiplication and inversion. In groups, compatibility can be characterized by a normal subgroup.
Quotient group : A group whose elements are cosets of a normal subgroup, with multiplication induced from the group. It is the group-theoretic quotient produced by a congruence.
Universal algebra : The study of algebraic structures defined by operations and equations. Congruence relations apply uniformly across the structures this field studies.
Quotient algebra : An algebraic structure formed from equivalence classes with operations induced from the original structure. Compatibility ensures operations on classes do not depend on chosen representatives.
Normal subgroup : A subgroup invariant under conjugation by every element of its group. Normal subgroups correspond exactly to congruences on groups.
Quotient ring : A ring of residue classes modulo an ideal, with induced addition and multiplication. Ring congruences provide precisely the identifications used to construct it.
Algebraic structure : A set equipped with operations satisfying specified laws. Congruence is defined relative to the operations of such a structure.
Kernel congruence : The congruence on an algebra induced by a homomorphism, relating elements with the same image. It shows how homomorphisms generate congruences through their fibers.
Ring congruence : An equivalence relation on a ring preserved by addition, additive inverse, and multiplication. Ring congruences correspond to ideals, linking relations with familiar algebraic structure.
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