Let $(P, L, I)$be a projective plane. For a line $l \in L$ let $p(l)= \{ p \in P | pIl \}$ be the points on $l$,and for a point $p \in P$ let $l(p)= \{ l \in L | pIl \}$ be the lines through $p$.
From: Jürgen Richter-Gebert (auth.) - Perspectives on Projective Geometry A Guided Tour Through Real and Complex Geometry,page-44
What does $pIl$ mean in this context?
$P$ and $L$ are arbitrary sets and their elements are considered as 'points' and 'lines', respectively.
$I$ is a relation between these sets, called the incidence relation (it contains certain pairs $(p,l)$ with $p\in P,\ l\in L$).
$p\,I\,l$ is the infix notation for expressing $(p,l)\in I$, which is interpreted as 'point $p$ lies on line $l$'.
$p(l)$ is the set of all points of line $l$ and $l(p)$ is the set of all lines through point $p$.
