Universal Algebra
The idea is generalized in universal algebra: A congruence relation on an algebra A is a subset of the direct product A × A that is both an equivalence relation on A and a subalgebra of A × A.
The kernel of a homomorphism is always a congruence. Indeed, every congruence arises as a kernel. For a given congruence ~ on A, the set A/~ of equivalence classes can be given the structure of an algebra in a natural fashion, the quotient algebra. The function that maps every element of A to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~.
The lattice Con(A) of all congruence relations on an algebra A is algebraic.
Read more about this topic: Congruence Relation
Famous quotes containing the words universal and/or algebra:
“Not because Socrates has said it, but because it is really in my nature, and perhaps a little more than it should be, I look upon all humans as my fellow-citizens, and would embrace a Pole as I would a Frenchman, subordinating this national tie to the common and universal one.”
—Michel de Montaigne (1533–1592)
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)