Usually, there are additional hypotheses on this structure. For instance, we often require that two lines are equal just in case they are incident to exactly the same points (and dually for points).
Geometrical algebra is the study of incidence geometries and the conditions under which they may be bi-interpretable with coordinatizing rings.
A pseudoplane is an incidence geometry for which any pair of distinct lines meets in a finite set of points and any pair of distinct points lies on only finitely many lines.