Congruence Relation - Universal Algebra

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:

    The world still wants its poet-priest, a reconciler, who shall not trifle with Shakspeare the player, nor shall grope in graves with Swedenborg the mourner; but who shall see, speak, and act, with equal inspiration. For knowledge will brighten the sunshine; right is more beautiful than private affection; and love is compatible with universal wisdom.
    Ralph Waldo Emerson (1803–1882)

    Poetry has become the higher algebra of metaphors.
    José Ortega Y Gasset (1883–1955)