Name: Anonymous 2007-09-28 10:00 ID:RdoOgSgT
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 homomorphic image 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 homomorphic image 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.