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.