Editing Incidence structure (section)

==Formal definition and terminology==
An '''incidence structure''' is a triple ({{math|''P'', ''L'', ''I''}}) where {{mvar|P}} is a set whose elements are called ''points'', {{mvar|L}} is a distinct set whose elements are called ''lines'' and {{math|''I'' ⊆ ''P'' × ''L''}} is the [[Incidence (geometry)|incidence]] [[heterogeneous relation|relation]]. The elements of {{mvar|I}} are called '''flags.''' If ({{math|''p'', ''l''}}) is in {{mvar|I}} then one may say that point {{mvar|p}} "lies on" line {{mvar|l}} or that the line {{mvar|l}} "passes through" point {{mvar|p}}. A more "symmetric" terminology, to reflect the [[Symmetry (mathematics)|symmetric]] nature of this relation, is that "{{mvar|p}} is ''incident'' with {{mvar|l}}" or that "{{mvar|l}} is incident with {{mvar|p}}" and uses the notation {{math|''p'' I ''l''}} synonymously with {{math|(''p'', ''l'') ∈ ''I''}}.<ref name=Demb1>{{harvnb|Dembowski|1968|pages=1–2}}</ref>

In some common situations {{mvar|L}} may be a set of subsets of {{mvar|P}} in which case incidence {{mvar|I}} will be containment ({{math|''p'' I ''l''}} if and only if {{mvar|p}} is a member of {{mvar|l}}). Incidence structures of this type are called ''set-theoretic''.<ref>{{harvnb|Biliotti|Jha|Johnson|2001|page=508}}</ref> This is not always the case, for example, if {{mvar|P}} is a set of vectors and {{mvar|L}} a set of [[square matrix|square matrices]], we may define 
<math display=block>I = \{(v, M) : \vec v \text{ is an eigenvector of matrix } M \}.</math>
This example also shows that while the geometric language of points and lines is used, the object types need not be these geometric objects.