1. ## definition if isomorphism?

What is the definition for ten years old of isomorphism...? What the definition of morphism?
I don't like Wikipedia.
Can you write in a single paragraph on the definition of them.

2. ## Re: definition if isomorphism?

Originally Posted by policer
What is the definition for ten years old of isomorphism...? What the definition of morphism?
I don't like Wikipedia. Can you write in a single paragraph on the definition of them.
You have asked an almost impossible question to answer for even high school. The reason being that there are several things going on at once here. Having two sets each has its own operation. If it is possible to match the elements of each one-to-one in such a way that operations preserved. That is very vary loose description.

3. ## Re: definition if isomorphism?

Thank you.
You can give an example. iN MY Understanding, it is a "type" of a unique function.

4. ## Re: definition if isomorphism?

Originally Posted by policer
Thank you.
You can give an example. iN MY Understanding, it is a "type" of a unique function.
You have given no context. Are you working with groups or rings or ideals or fields or what?

5. ## Re: definition if isomorphism?

Nobody here is in the business of giving you answers that make you appear to be gifted.

How about you start by expressing your own understanding, perhaps with an example.

6. ## Re: definition if isomorphism?

OK.
I knew that all the "creators" you give me is for establishing theories in number theory. [Field and group, and etc.] and also my student know.
I write that my explanation is give to smart [or my Enlglish is bad by the word "gifted", it can be an error in using of the English language.]
My student know what is field and what is ring.
Can This help me to explain them what is isomorphism?!

7. ## Re: definition if isomorphism?

An isomorphism from one algebraic structure to another is a "one-to-one", "onto" mapping, f, that "preserves" operations. That is, if x+ y is defined in the first algebraic structure then f(x)+ f(y) must be defined in the second algebraic structure and f(x+ y)= f(x)+ f(y). If x o y is an operation defined in the first algebraic structure then f(x) o f(y) must be defined in the second structure and f(x o y)= f(x) o f(y).

Another way of looking at it is that two algebraic structures are isomorphic if and only if the second is just the first with things "renamed".

8. ## Re: definition if isomorphism?

Thanks, that what I want.

9. ## Re: definition if isomorphism?

Axioms of E: Is axioms of E functions is also isomorphism? (That what my student ask me)?

10. ## Re: definition if isomorphism?

Here is what you posted,
Originally Posted by policer
What is the definition for ten years old of isomorphic...? What the definition of morphism?
I don't like Wikipedia. Can you write in a single paragraph on the definition of them.
Thus I posted a greatly dumb-down version of the Wikipedia piece.
Now you post this hard to understand request.
Originally Posted by policer
Axioms of E: Is axioms of E functions is also isomorphism? (That what my student ask me)?
What does Axioms of E mean? Look at Wikipedia again.
There is a discussion of the mapping $\exp:\mathbb{R}\to\mathbb{R}^+$ given by $x\mapsto e^x$.
You see that $\exp(xy)=\exp(x)\exp(y)$

11. ## Re: definition if isomorphism?

What the different between isomorphism to homomorphism?
If you can write in a little line.
Thank you for everybody, you are very helpful.
I need to give only a conclusion of the matter. So, it is not matter so much.
I ask for this example only.

12. ## Re: definition if isomorphism?

Originally Posted by policer
What the different between isomorphism to homomorphism?
If you can write in a little line.
Thank you for everybody, you are very helpful.
An isomorphism between algebraic structures of the same type is commonly defined as a bijective homomorphism.