Name: Anonymous 2007-09-28 9:59 ID:D+umMyO5
We classify morphisms according to their properties as functions. If Φ : A → A' is a morphism, then we call Φ an isomorphism if Φ is bijective; an epimorphism if Φ is surjective, and a monomorphism if Φ is injective.
Some further characterisations: The abstract properties of an algebraic system are exactly those whch are invariant (i.e., which do not change) under isomorphism. For epimorphisms, A' is called the [i]homomorphic image[i] of A, and we regard (A', Ω') as an abstraction or a model of (A, Ω). A monomorphism A → A' is sometimes called an embedding of A into A'.
We single out morphisms that map algebras onto themselves. We called a morphism Φ : A → A' that maps (A, Ω) onto itself an endomorphism. If Φ is also bijective, hence an isomorphism, Φ : A → A, then we call it an automorphism.
Some further characterisations: The abstract properties of an algebraic system are exactly those whch are invariant (i.e., which do not change) under isomorphism. For epimorphisms, A' is called the [i]homomorphic image[i] of A, and we regard (A', Ω') as an abstraction or a model of (A, Ω). A monomorphism A → A' is sometimes called an embedding of A into A'.
We single out morphisms that map algebras onto themselves. We called a morphism Φ : A → A' that maps (A, Ω) onto itself an endomorphism. If Φ is also bijective, hence an isomorphism, Φ : A → A, then we call it an automorphism.