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Please explain the different morphisms to me

  1. Nov 20, 2008 #1
    Can someone please explain the differences between:
    An Automorphism
    An Isomorphism
    A Homomorphism
    and An Endomorphism
     
  2. jcsd
  3. Nov 20, 2008 #2
    These terms are unrelated. The suffix morphism, in general, used for functions with special properties. There's also homeomorphisms, diffeomorphisms, and a slew of others, but they don't necessarily relate to each other.

    But anyway. I'm not sure exactly about automorphisms. They deal with groups and symmetry, I believe, but I haven't done much with group theory.

    Homomorphisms are very important in group theory. If you have two groups (G, *) and (G', +), f is a homomorphism iff f(x * y) = f(x) + f(y). An example would be the exponential function f(x) = e^x between the groups (R, *) and (R, +), because e^(xy) = e^x + e^y.

    'Isomorphism' can have multiple meanings in different areas. In group theory, it's a bijection (a one-to-one and onto) function which is a homomorphism and whose inverse is also a homomorphism. So f(x*y) = f(x) + f(y) and f^-1(x') + f^-1(y') = f^-1(x' * y'). In category theory, it's a homomorphism over the composition operator.

    I'm not sure about endomorphisms either. They are part of category theory again.
     
  4. Nov 20, 2008 #3

    HallsofIvy

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    You could just look them up. An isomorphism (from the Greek for "same", "change") from one algebraic structure to another is a one-to-one, onto function that "preserves" all operations: f(x+ y)= f(x)+ f(y), f(xy)= f(x)f(y) if both addition and multiplicaton are defined in the two structures.

    An automorphism (from the Greek for "self", "change") is just an isomorphism from an algebraic structure to itself.

    A homomorphism (again "same", "change") is a function from one algebraic structure to another that preserves the operations- but is not necessarily one-to-one or onto.

    An endomorphism (from the Greek for "inside", "change") is a homomorphism from an algebraic structure to itself.
     
  5. Nov 20, 2008 #4
  6. Nov 20, 2008 #5

    jambaugh

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    Let's see, a Homomorphism is a mapping preserving some structural relation, usually a product. Thus H(a)H(b)=H(ab).

    An endomorphism is a homomorphism from a object into itself as opposed to say into another object.

    An isomorphism is an invertible homomorphism and thus is one-to-one or bijective.
    An automorphism is both endomorphism and isomorphism.

    Example: Consider the set of complex numbers under addition.

    An automorphism would be to map [tex]z \mapsto -z[/tex].
    An endomorphism would be to map [tex]z \mapsto i\cdot\Re(z)[/tex].
    Another endomorphism would be to map [tex]z \mapsto 0[/tex].

    A homomorphism would be to map [tex]z \mapsto e^{zA}[/tex] where A is some square matrix and you are considering the set of invertible matrices under the operation of multiplication.
    This homomorphism is also I believe an isomorphism.

    A non-isomoprhic homomorphism would be to map [tex]z\mapsto I[/tex], the identity matrix under this same group of invertible matrices with multiplication. This is actually the composition of the zero map above with the previous matrix map.
     
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