Binary Structures and Mapping

I am having great difficulty conceptualizing binary structures and mapping from one to the other.

Here's my problem:

The map $\displaystyle \phi : \mathbb{Z} \rightarrow \mathbb{Z}$ defined by $\displaystyle \phi(n) = n + 1$ for $\displaystyle n \in \mathbb{Z}$ is one-to-one and onto $\displaystyle \mathbb{Z}$. Give the definition of a binary operation $\displaystyle *$ on $\displaystyle \mathbb{Z}$ such that $\displaystyle \phi$ is an isomorphism mapping.

$\displaystyle <\mathbb{Z},\cdot> with <\mathbb{Z},*>$.

So I have $\displaystyle <\mathbb{Z},\cdot>$. This means that for members a and b, this structure is $\displaystyle a \cdot b$ and $\displaystyle d \cdot a$, right? And I'm trying to map this to an unknown structure? I don't understand what exactly I'm doing.

What is that $\displaystyle \phi(n) = n + 1$ function? I assume it takes a number in $\displaystyle \mathbb{Z}$ and maps it to a number in $\displaystyle \mathbb{Z}$, specifically one more than the input number. But what does that have to do with my structures? What am I trying to find?

In utter confusion, I took $\displaystyle <\mathbb{Z},\cdot>$ and figured that for the inputs of a=2 and b=3, the answer would be 6 since 2*3=6. So I take that 6... add one to it from $\displaystyle \phi$, and that gives me 7... is that 7 the value that is supposed to correspond with the second structure, $\displaystyle <\mathbb{Z},*>$? So is my task to find a function $\displaystyle *$ that takes two variables $\displaystyle a$ and $\displaystyle b$ to make 7, or more abstractly, ab+1?? (Headbang)

Any help = appreciated.

Re: Binary Structures and Mapping

what you are being asked to do is FIND a definition for * so that φ(ab) = φ(a)*φ(b).

the fact that φ is 1-1 and onto already is the "iso" part....ensuring that φ(ab) = φ(a)*φ(b) is the "morphism" part.

suppose that a*b is defined as ab+1...does this definition of * fit the requirements?

(your question is a little vague...since no other information is given, i am assuming you aren't requiring that * be associative, or have an identity).