I am looking at a graph theory problem that describes the partite sets of a bipartite as two copies of the $(m+1)$-dimensional vector space over the finite field $\mathbb{F}_{p^n}$ ($p$ is prime and $n\geq 1$ is an integer), $P$ and $L$. They call the elements of $P$ "points" and the elements of $L$ "lines".
They say a point $(p)=(p_1, \ldots, p_{m+1})\in P$ is adjacent to a line $[l]=[l_1, \ldots, l_{m+1}]$ if and only if
$$ l_{i+1}+p_{i+1}=l_ip_1 $$
for every $i\in \left\{1,2,\ldots, m \right\}$.
I would like to know what the sum on the left hand side and the product on the right side mean geometrically in terms of lines and points. Perhaps I should ignore the geometric meaning? I just want to know how I should be viewing the above definition.
Thank you!
There is little geometric meaning in the addition and multiplication - because the formula is used to define the very incidence structure of points and lines of this example structure.
Instead, in order to justify the denotion as "points" and "lines", you should verify that the arithmetic properties of $+$ and $\times$ can be used to show that this incidence structure satisfies the axioms of geometry. For example, if $p,q$ are distinct points, then we want to show that there exist one and only one $l$ such that both points are incident with that line, i.e., the system of equations $$\begin{align}p_{i+1}+x_{i+1}&=p_1x_i,\\q_{i+1}+x_{i+1}&=q_1x_i\end{align}$$ has exactly once solution.