List the things f has to do to be a homomorphism

Bachelier

The function

##f(x) = \frac{x}{x+1} ##

is not a Homomorphism because f(1) ≠ 1..Am I correct?

Simon Bridge

List the things f has to do to be a homomorphism.

Bachelier

Assume we are in ℝ then

f(ab) = f(a).f(b) under multiplication
f(a+b) = f(a)+f(b) under addition

and that's all I find in my book. but I know we need to check some extra stuff.

Simon Bridge

OK - but if it fails any one of the conditions, then it isn't a homomorphism right?
Have you applied either of those two tests to this situation?

Your example was f(1)=1 - proposed as a test.
How does that work in with the relations you listed?
In the first, ab=1 and in the second a+b=1.

You could also consider what sort of transformation is represented by f(x) ... i.e. is f(x) defined for all real x? Does it have to be if it is to be a homomorphism?

Jim Kata

Are you sure that you don't mean homeomorphism, not homomorphism?

Jim Kata

I'll assume you mean homeomorphism then f(x) must be continuous and have a continuous inverse f(x) is not continuous at x = -1 and the inverse function is not continuous at f^-1=1.

micromass

Could you please always list what structure you are working with. Saying that "f is a homomorphism" is a meaningless statement. You should state "f is a homomorphism of groups/rings/fields/algebras/lattices/..."

Also, be sure to always give the domain and codomain.

Bachelier

Homomorphism of groups. mainly define our f: [0,∞) → ℝ.

I swear sometimes I just need some sleep. of course it is not because it fails property 1. mainly f(a+b) ≠ f(a) + f(b).
Sometimes my brain will jump to the most complex property and try to solve it while ignoring the simplest ones.

That aside, a Homomorphism of groups must send the identity element of the domain to the identity element of the codomain, right?

Bachelier

If I define my function f from ℝ to ℝ then the function is not defined at x = -1.

I take it from your question that a Homomorphism of groups must be well-defined on all elements of the group?

micromass

OK, but [itex][0,\infty)[/itex] is not a group. So you can't talk about homomorphism of groups. Furthermore, a group only has one operation. So, saying that a homomorphisms of groups satisfy

[tex]f(x+y)=f(x)+f(y)~\text{and}~f(xy)=f(x)f(y)[/tex]

is not correct. Why not? Because now you're talking about two operations: addition and multiplication. A group is a set with only one operation (which satisfies some conditions.

So if you have a function [itex]f:(\mathbb{R},+)\rightarrow (\mathbb{R},+)[/itex] (I usually denote a group by [itex](G,*)[/itex], where G is a set and * is an operation on the set), then this is a homomorphism if and only if [itex]f(x+y)=f(x)+f(y)[/itex]. The multiplication has nothing to do with this.

In general, a function [itex]f:(G,*)\rightarrow (H,\oplus)[/itex] must satisfy [itex]f(x*y)=f(x)\oplus f(y)[/itex]. Nothing more.

If you want to talk about two operations (like addition and multiplication on [itex]\mathbb{R}[/itex]), then you have to talk about rings.

Yes. But what you mean with identity element depends on the group operation. In the group [itex](\mathbb{R},+)[/itex], the identity is 0. In the groups [itex](\mathbb{R}\setminus\{0\},\cdot)[/itex], the identity is 1.

micromass

Yes. But that's not only true for homomorphisms. It is in general true for functions. A function [itex]f:X\rightarrow Y[/itex] must be defined on all [itex]x\in X[/itex].

So [itex]f:\mathbb{R}\rightarrow \mathbb{R}:x\rightarrow \frac{1}{x}[/itex] is not a function (because not defined in 0). But [itex]f:\mathbb{R}\setminus \{0\}\rightarrow \mathbb{R}:x\rightarrow \frac{1}{x}[/itex] is a function.

Bachelier

Beautiful Math and very clear definitions..Thank you very much.

[0,∞) is indeed not a group because of the inverse axiom. For instance wrt multiplication 0 has no inverse. (if the operation is addition, then no element has an inverse)

So I guess I can only call [0,∞) an interval or a set.

Now ##((0,∞), *)## where * is the regular multiplication is a group under ##*##, right?

micromass

Yes, it is. You may be surprised to learn that [itex]((0,+\infty),\cdot)[/itex] is actually isomorphic (as group) to [itex](\mathbb{R},+)[/itex].

Indeed, the isomorphism is

[tex]T:(\mathbb{R},+)\rightarrow ((0,+\infty),\cdot):x\rightarrow e^x.[/tex]

Bachelier

After learning a long time ago that ##(0,1) \cong \mathbb{R}##, nothing surprises me anymore... lol :)

micromass

OK, but that's not as groups.