Incidence Structures and Incidence Matrices

The most basic notion in (finite) geometry is that of an incidence structure. It contains nothing more than the idea that two objects from distinct classes of things (say points and lines) may be “incident” with each other. The only requirement will be that the classes do not overlap. We now make this more precise.

Definitions.

An incidence structure is a triple D = (V, B, I) where V and B are any two disjoint sets and I is a binary relation between V and B, i.e. I ⊆ V × B. The elements of V will be called points, those of B blocks and those of I flags. Instead of (p, B) ∈ I, we will simply write pI B and use such geometric language as “the point p lies on the block B”, “B passes through p”, “p and B are incident”, etc.

For reasons of convenience, we will usually not state whether a given object is a point or a block; this will be clear from the context and we will always use lower case letters (e.g. p, q, r, …) to denote points and upper case letters (e.g. B, C, …) to denote blocks. Now let us look at some examples! Of course, familiar (euclidean) geometry provides examples, e.g. taking points and lines (as blocks) in the euclidean plane or points and planes (as blocks) in 3-space. One reason for choosing the term “block” instead of “line” is that we will very often consider planes or hyperplanes as blocks.