Relationship To Algebraic Kernels
Universal algebra defines a notion of kernel for homomorphisms between two algebraic structures of the same kind. This concept of kernel measures how far the given homomorphism is from being injective. There is some overlap between this algebraic notion and the categorical notion of kernel since both generalize the situation of groups and modules mentioned above. In general, however, the universal-algebraic notion of kernel is more like the category-theoretic concept of kernel pair. In particular, kernel pairs can be used to interpret kernels in monoid theory or ring theory in category-theoretic terms.
Read more about this topic: Kernel (category Theory)
Famous quotes containing the words relationship and/or algebraic:
“The relationship between mother and professional has not been a partnership in which both work together on behalf of the child, in which the expert helps the mother achieve her own goals for her child. Instead, professionals often behave as if they alone are advocates for the child; as if they are the guardians of the child’s needs; as if the mother left to her own devices will surely damage the child and only the professional can rescue him.”
—Elaine Heffner (20th century)
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)