I am teaching myself from a math paper ("Explicit construction of graphs with an arbitrary large girth and of large size" by Felix Lazebnik, and Vasiliy A. Ustirnenko) and I have questions about the way they they defined the vectors of the incidence structure. I weaved my questions into the text, so I bolded all of my questions to make them easier to find.
Let q be a prime power. The infinite semiplane $\Gamma$(q) is defined as follows: Let P and L be two infinite-dimensional vector spaces over the finite field $\mathbb{F}_q$ . The vectors of P and L can be thought as infinite sequences of elements of $\mathbb{F}_q$. P and L will be the set of points and the set of lines of the incidence structure $\Gamma$(q) [This confused me a little because $\Gamma$(q) was defined to be an infinite semiplane, is that the same as an incidence structure?]. A vector p$\epsilon$P will be denoted by (p), and a vector l$\epsilon$L by [l]. The parentheses and brackets will allow us to distinguish vectors of different types (points and lines).
Up until this point in the paper, I'm pretty good and I kind of get what's going on. But then the following is where I'm really confused:
It will be convenient for us to write the components of points and lines as: $$(p) = (p_1, p_{1,1}, p_{1,2}, p_{2,2}, p_{2,2}', p_{2,3}, p_{3,2}, p_{3,3}, p_{3,3}', ... , p_{i,i}', p_{i+1,i}, p_{i,i+1}, p_{i+1,i+1}, ... , )$$ $$[l] = [l_1, l_{1,1}, l_{1,2}, l_{2,2}, l_{2,2}', l_{2,3}, l_{3,2}, l_{3,3}, l_{3,3}', ... , l_{i,i}', l_{i+1,i}, l_{i,i+1}, l_{i+1,i+1}, ... , ]$$
Why are some 'prime' (What is the difference between $p_{2,2}$ and $p_{2,2}'$)? Can the indices only differ by 1? Can $l_{1,4}$ exist? Can $p_{23,7}$ exist? The paper goes on to say:
We also assume $p_{-1,0} = l_{0,-1} = p_{1,0} = l_{0,1} = 0$ ; $p_{0,0} = l_{0,0} = - 1$ ; $p_{0,0}' = l_{0,0}' = 1$ ; $p_{0,1} = p_1$ ; $l_{1,0} = l_{1}$ ; $l_{1,1}' = l_{1,1}$, $p_{1,1}' = p_{1,1}$
All of a sudden the indices can be negative, what does that mean? Is -1 the only possible negative index? And then my final question comes from the next portion:
We say that a point (p) is incident with a line [l], and write it as (p)I[l] if and only if the following conditions are satisfied for i=1,2 .... :
$$ l_{i,i} - p_{i,i} = l_1p_{i-1,i} \\ l_{i,i}' - p_{i,i}' = p_1p_{i,i-1} \\ l_{i,i+1} - p_{i,i+1} = p_1l_{i,i} \\ l_{i+1,i} - p_{i+1,i} = l_1p_{i,i}' $$
I'm assuming the subtraction is straightforward, and that the multiplication is a cross product (if I'm wrong, feel free to let me know), but I'm not understanding how these conditions imply that a point (p) will be on a line [l]. Can anyone clarify that?
Any help in anyway would be greatly appreciated.