Algebras With Nonalgebraic Structure
Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations. For example, one may consider topological groups or topological vector spaces, with are equipped with a topology. In this case, we would expect the homomorphism f to preserve this additional structure; in the topological examples, we would want f to be a continuous map. The process may run into a snag with the quotient algebras, which may not be well-behaved. In the topological examples, we can avoid problems by requiring that topological algebraic structures be Hausdorff (as is usually done); then the kernel (however it is constructed) will be a closed set and the quotient space will work fine (and also be Hausdorff).
Read more about this topic: Kernel (algebra)
Famous quotes containing the word structure:
“The question is still asked of women: “How do you propose to answer the need for child care?” That is an obvious attempt to structure conflict in the old terms. The questions are rather: “If we as a human community want children, how does the total society propose to provide for them?””
—Jean Baker Miller (20th century)