An affine plane of order $n$ has $n^2$ points, $n$ points per line, $n+1$ lines intersecting each point, every two points on one line, and $n^2+n$ lines. The plane can be partitioned into $n+1$ classes, each with $n$ parallel lines in it.
For Desarguesian planes, it is possible to represent the $n+1$ classes with the equations for lines, where multiplication and addition are in $GF(n)$:
$y=mx+b$
and $x=b$
where there are $n$ choices for $m$ and $n$ choices for $b$.
For non-Desarguesian planes, we cannot do this in $GF(n)$, but is it still possible to do something like this, so that there is still some nice algebraic structure left?