of the graph form the line set, and incidence means that a vertex is an endpoint of an edge. Linear spaces Incidence structures are seldom studied in their full generality; it is typical to study incidence structures that satisfy some additional axioms. For instance, a partial linear space is an incidence structure that satisfies: Any two distinct points are incident with at most one common line, and Every line is incident with at least two points. If the first axiom above is replaced by the stronger: Any two distinct points are incident with exactly one common line, the incidence structure is called a linear space. Nets A more specialized example is a k-net. This is an incidence structure in which the lines fall into k parallel classes, so that two lines in the same parallel class have no common points, but two lines in different classes have exactly one common point, and each point belongs to exactly one line from each parallel class. An example of a k-net is the set of points of an affine plane together with k parallel classes of affine lines. Dual structure If we interchange the role of "points" and "lines" in we obtain the dual structure, where is the converse relation of . It follows immediately from the definition that: This is an abstract version of projective duality. A structure that is isomorphic to its dual is called self-dual. The Fano plane above is a self-dual incidence structure. Other terminology The concept of an incidence structure is very simple and has arisen in several disciplines, each introducing its own vocabulary and specifying the types of questions that are typically asked about these structures. Incidence structures use a geometric terminology, but in graph theoretic terms they are called hypergraphs and in design theoretic terms they are called block designs. They are also known as a set system or family of sets in a general context. Hypergraphs Each hypergraph or set system can be regarded as an incidence structure in which the universal set plays the role of "points", the corresponding family of sets plays the role of "lines" and the incidence relation is set membership "∈". Conversely, every incidence structure can be viewed as a hypergraph by identifying the lines with the sets of points that are incident with them. Block designs A (general) block design is a set together with a family of subsets