= Incidence structure
{title2=$(P,B,I),\quad I\subseteq P\times B$}
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> is
$$
M=\begin{pmatrix}1&0&1\\1&1&0\\0&1&1\end{pmatrix}.
$$
Every 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.<FinInG manual>[section 4.1, Incidence structures]{ref}
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