Is there any order behind the chaos of multiplication modulo $p$? Let's lift the finite group $\mathbb{Z}_p$ to $\mathbb{Z}$ when we're taking magnitudes so the usual ordering we're using.  So we say $a < b$ in $\mathbb{Z}_p$ iff $a < b$ under inclusion into $\mathbb{Z}$.  That is all unimportant with respect to my question but in case anyone was wondering what ordering we refer to.  
So we know that, modulo $p$, $a \lt b$ does not imply $ca \lt cb$, but has anyone studied what actually happens to the direction of $\lt$ under multiplication?
It would be of use to me for googling if I knew what to call a group together with an ordering on the group since google confuses this order with "order of an element".
 A: The quick and short aswer is that there is no useful ordering in $\Bbb{Z}_p$. A fundamental difficulty is that the "lift" is not unique. Modulo $11$ do you want to lift $10$ as $10$ or $-1$, or $-12\ldots$? Yes, we can make lifting well-defined by insisting
that the lifts are in the interval $[0,p-1]$ or some fixed interval of $p$ integers. Another popular choice is $[-(p-1)/2,(p-1)/2]$. Defining the lifting is not the problem. Lack of an order relation compatible with arithmetic is.
This means that people who are used to solving equations by methods that depend on having the order relation around (such as finding a solution of an equation by dissecting smaller and smaller intervals) need to learn other methods. Lack of ordering means lack of concept of a "small error", so a method based on improving an approximate solution loses its basis.
Yet, ordering the lifts is occasionally useful. Gauss' Lemma comes to mind. There we are to count how many of the lifts of $a,2a,\ldots,\dfrac{(p-1)}2a$ are larger than $p/2$. That leads eventually to the law of quadratic reciprocity. An essential tool in deciding whether quadratic equations have a solution modulo $p$ or not (among other things).
Lack of an ordering occasionally gives a headache to people working in continuous math side also. Angles between two lines have a similar ambiguity coming from the fact an angle is often defined only up to an integer multiple of $2\pi$. Usually the difficulties there are not severe, because confining ourselves to an interval of length $2\pi$ allows us to isolate any eventual discontinuities, and we can e.g. integrate using polar coordinates quite well. The type of problems that linger are of the type: what's the average of two angles? Is the average of $+1^\circ$ and $-1^\circ$ zero? Why does it change, when we use $359^\circ$ in place of $-1^\circ$? Of course, in a sense, all of the calculus of residues in complex analysis rides on the non-uniqueness of the lift of a compass direction to a real number :-)
