Finite Geometry

Definition

A finite geometry (of rank $2$) is a triple $\mathcal{S}=(\mathcal{P},\mathcal{L},\mathrm{I})$, where $\mathcal{P}$, $\mathcal{L}$ are disjoint non-empty finite sets and $\mathrm{I}\subseteq \mathcal{P}\times \mathcal{L}$ is a relation, the incidence relation.

The dual of $\mathcal{S}$ is the geometry ${\mathcal{S}}^D=(\mathcal{L},\mathcal{P},{\mathrm{I}}^D)$ with $L{\mathrm{I}}^{D}P⟺P\mathrm{I}L$.

A flag of $\mathcal{S}$ is a pair $\{P,L\}$ with $P\mathrm{I}L$, and an anti-flag is a pair $\{P,L\}$ such that $P,L$ are not incident.

Two points $P,Q$ (lines $L,M$) are collinear (concurrent) if they are incident with at least one comment line (point), denoted by $P\sim Q$ ($L\sim M$).

A sub-geometry of $\mathcal{S}$ is a geometry ${\mathcal{S}}^{\prime }=({\mathcal{P}}^{\prime },{\mathcal{L}}^{\prime },{\mathrm{I}}^{\prime })$ with ${\mathcal{P}}^{\prime }\subseteq P$, ${\mathcal{L}}^{\prime }\subseteq \mathcal{L}$ and ${\mathrm{I}}^{\prime }=\mathrm{I}\cap ({\mathcal{P}}^{\prime }\times {\mathcal{L}}^{\prime })$.