Please explain the different morphisms to me

[Diffy]

Can someone please explain the differences between:
An Automorphism
An Isomorphism
A Homomorphism
and An Endomorphism

 

[Tac-Tics]

Can someone please explain the differences between:
An Automorphism
An Isomorphism
A Homomorphism
and An Endomorphism

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.

 

[HallsofIvy]

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.

 

[Tac-Tics]

Ack. I guess these terms aren't as unrelated as I thought X-(

http://en.wikipedia.org/wiki/Endomorphism

There's a graph right there with links to all the other pages.

 

[jambaugh]

Can someone please explain the differences between:
An Automorphism
An Isomorphism
A Homomorphism
and An Endomorphism

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 z \mapsto -z.
An endomorphism would be to map z \mapsto i\cdot\Re(z).
Another endomorphism would be to map z \mapsto 0.

A homomorphism would be to map z \mapsto e^{zA} 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 z\mapsto I, 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.