Relation With Homomorphisms
If ƒ: A → B is a homomorphism between two algebraic structures (such as homomorphism of groups, or a linear map between vector spaces), then the relation ≡ defined by
- a1 ≡ a2 if and only if ƒ(a1) = ƒ(a2)
is a congruence relation. By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphic to the quotient of A by this congruence.
Read more about this topic: Congruence Relation
Famous quotes containing the word relation:
“There is undoubtedly something religious about it: everyone believes that they are special, that they are chosen, that they have a special relation with fate. Here is the test: you turn over card after card to see in which way that is true. If you can defy the odds, you may be saved. And when you are cleaned out, the last penny gone, you are enlightened at last, free perhaps, exhilarated like an ascetic by the falling away of the material world.”
—Andrei Codrescu (b. 1947)