Incidence structure

Remarks

Example: let $P = \set{1, 2, 3}$, let $L = \set{a, b}$, and let $I = \set{(1, a), (2, a), (2, b), (3, b)}$. Check: $P$ and $L$ are disjoint and $I$ is a subset of $P \times L$, so $(P, L, I)$ is an incidence structure, a path of two lines meeting at the point $2$.

Non-example: with the same $P$ and $L$, a set $I$ that contains the pair $(1, 2)$ is not an incidence relation, because $(1, 2)$ is not in $P \times L$.

The dual of $(P, L, I)$ is $(L, P, I^{\top})$, where $(\ell, p) \in I^{\top}$ exactly when $(p, \ell) \in I$. An isomorphism from $(P, L, I)$ to $(P', L', I')$ is a pair of bijections $P \to P'$ and $L \to L'$ that preserves incidence in both directions. Often lines are taken to be subsets of $P$, with incidence given by membership. The general definition also allows two different lines with the same set of points. A finite incidence structure is described by its incidence matrix.