Incidence structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the relation of which points are incident on which lines for all points and lines. What is left is the incidence structure of the Euclidean plane.
Incidence structures are most often considered in the geometrical context where they are abstracted from, and hence generalize, planes, but the concept is very broad and not limited to geometric settings. Even in a geometric setting, incidence structures are not limited to just points and lines; higher-dimensional objects can be used. The study of finite structures is sometimes called finite geometry.
Formal definition and terminology
An incidence structure is a triple where is a set whose elements are called points, is a distinct set whose elements are called lines and is the incidence relation. The elements of are called flags. If is in then one may say that point "lies on" line or that the line "passes through" point. A more "symmetric" terminology, to reflect the symmetric nature of this relation, is that " is incident with " or that " is incident with " and uses the notation synonymously with.In some common situations may be a set of subsets of in which case incidence will be containment. Incidence structures of this type are called set-theoretic. This is not always the case, for example, if is a set of vectors and a set of square matrices, we may define
This example also shows that while the geometric language of points and lines is used, the object types need not be these geometric objects.
Examples
An incidence structure is uniform if each line is incident with the same number of points. Each of these examples, except the second, is uniform with three points per line.Graphs
Any graph is a uniform incidence structure with two points per line. For these examples, the vertices of the graph form the point set, the edges 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.
- Any two distinct points are incident with exactly one common line,
Nets
A more specialized example is a -net. This is an incidence structure in which the lines fall into 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 -net is the set of points of an affine plane together with parallel classes of affine lines.Dual structure
If we interchange the role of "points" and "lines" inwe 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 incidencestructure in which the universal set plays the role of "points", the corresponding family of subsets 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 block design is a set together with a family of subsets of . Normally a block design is required to satisfy numerical regularity conditions. As an incidence structure, is the set of points and is the set of lines, usually called blocks in this context. If all the subsets in have the same size, the block design is called uniform. If each element of appears in the same number of subsets, the block design is said to be regular. The dual of a uniform design is a regular design and vice versa.Example: Fano plane
Consider the block design/hypergraph given by:This incidence structure is called the Fano plane. As a block design it is both uniform and regular.
In the labeling given, the lines are precisely the subsets of the points that consist of three points whose labels add up to zero using nim addition. Alternatively, each number, when written in binary, can be identified with a non-zero vector of length three over the binary field. Three vectors that generate a subspace form a line; in this case, that is equivalent to their vector sum being the zero vector.
Representations
Incidence structures may be represented in many ways. If the sets and are finite these representations can compactly encode all the relevant information concerning the structure.Incidence matrix
The incidence matrix of a incidence structure is a matrix that has its rows indexed by the points and columns indexed by the lines where the -th entry is a 1 if and 0 otherwise. An incidence matrix is not uniquely determined since it depends upon the arbitrary ordering of the points and the lines.The non-uniform incidence structure pictured above is given by:
An incidence matrix for this structure is:
which corresponds to the incidence table:
| 0 | 0 | 0 | 1 | 1 | 0 | |
| 0 | 0 | 0 | 0 | 1 | 1 | |
| 1 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 1 | 0 | 0 | 0 | |
| 1 | 0 | 0 | 0 | 0 | 0 | |
| 1 | 1 | 1 | 1 | 0 | 1 |
If an incidence structure has an incidence matrix, then the dual structure has the transpose matrix T as its incidence matrix.
An incidence structure is self-dual if there exists an ordering of the points and lines so that the incidence matrix constructed with that ordering is a symmetric matrix.
With the labels as given in example number 1 above and with points ordered and lines ordered, the Fano plane has the incidence matrix:
Since this is a symmetric matrix, the Fano plane is a self-dual incidence structure.