Incidence structure ($(P,B,I),I\subseteq P\times B$)

In the point-block convention, an incidence structure consists of a set $P$ of points, a set $B$ of blocks, and a relation $I\subseteq P\times B$ specifying which points lie on which blocks. Blocks are abstract objects: distinct blocks may have identical sets of incident points unless a simplicity condition forbids this. A finite incidence structure is encoded by its incidence matrix of a set system, with entry $1$ for incidence and $0$ otherwise.

For example, take points $1,2,3$ and three blocks incident with $\{1,2\}$, $\{2,3\}$ and $\{1,3\}$. With points indexing rows, the incidence matrix of a set system isEvery point lies on two blocks, every block contains two points, and every pair of distinct points lies on exactly one block. It is a $2$-$(3,2,1)$ combinatorial design. Its matrix rank is three over $\mathbb{Q}$ but two over ${\mathbb{F}}_2$, since its determinant is two. This illustrates why incidence and the coefficient field jointly matter in algebraic counting arguments.

Finite geometries, graphs and combinatorial designs all supply incidence structures, but additional axioms distinguish their particular classes. Merely specifying incidence does not require any balanced-design or projective-plane axioms. Typed incidence structures can include further element types, such as points, lines and planes.^[ref]