Partial geometry

An incidence structure ${\displaystyle C=(P,L,I)}$ consists of points ${\displaystyle P}$, lines ${\displaystyle L}$, and flags ${\displaystyle I\subseteq P\times L}$ where a point ${\displaystyle p}$ is said to be incident with a line ${\displaystyle l}$ if ${\displaystyle (p,l)\in I}$. It is a (finite) partial geometry if there are integers ${\displaystyle s,t,\alpha \geq 1}$ such that:

• For any pair of distinct points ${\displaystyle p}$ and ${\displaystyle q}$, there is at most one line incident with both of them.
• Each line is incident with ${\displaystyle s+1}$ points.
• Each point is incident with ${\displaystyle t+1}$ lines.
• If a point ${\displaystyle p}$ and a line ${\displaystyle l}$ are not incident, there are exactly ${\displaystyle \alpha }$ pairs ${\displaystyle (q,m)\in I}$, such that ${\displaystyle p}$ is incident with ${\displaystyle m}$ and ${\displaystyle q}$ is incident with ${\displaystyle l}$.

A partial geometry with these parameters is denoted by ${\displaystyle \mathrm {pg} (s,t,\alpha )}$.