In modular arithmetic is the concept of "increasing" well defined? If there is a function $f(x) = x\bmod n$ and, whenever $0\leq x_1\lt x_2\lt n$, we have $f(x_1)\lt f(x_2)$, can we say that $f$ is increasing? 
Also, when finally I prove that $f(x)$ is increasing, how do I say it is doing so "modular arithmetically". I want to be able to specify that the function is increasing by the definition of increasing functions that is used for modular arithmetic. 
Disclosure: I'm not a mathematician: I'm a student programmer who loves math. I'm working on proving that a given algorithm satisfies the requirements for being a solution to the critical section problem. This is homework, but the answer to that question is not homework.
Thanks!
z. 
PS I post on stackoverflow but this is my first question here. 
 A: Mathematicians don't usually put orderings on rings with positive characteristic, since we like to be able to say things like $x<x+1$ for all $x$.
But there's no reason you can't order the residue classes by setting $[x]<[y]$ whenever $0\leq x,y<n$. In other words, $[0]<[1]<\ldots <[n-1]$. Be careful though. For example, if this is your ordering mod 5, then $[-1]>[12]$ since $[-1]=[4]$ and $[12]=[2]$.
Just make sure that your ordering does not depend on which representative you pick from each class. (This is the definition of well-defined.)
As far as terminology goes, this is a nonstandard thing, so it's worth explaining your ordering. Then you can say "$f$ is increasing with respect to this ordering."
A: If you want to have $x+1\gt x$ for all $x$, then there is no way to order the integers modulo $n$. 
If you are willing to settle for $0\lt1\lt2\lt\cdots\lt n-1$, and you have a function with $x\lt y$ implying $f(x)\lt f(y)$, then your function can only be the identity function, $f(x)=x$ for all $x$. 
