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==Incidence structures==

==Incidence structures==

The ''incidence matrix'' of an [[incidence structure]] ''C'' is a {{nowrap|''p'' × ''q''}} matrix ''B'' (or its transpose), where ''p'' and ''q'' are the number of ''points'' and ''lines'' respectively, such that {{nowrap|1=''B''_''i'',''j'' = 1}} if the point ''p''_i and line ''L''_''j'' are incident and 0 otherwise. In this case, the incidence matrix is also a [[biadjacency matrix]] of the [[Levi graph]] of the structure. As there is a [[hypergraph]] for every Levi graph, and ''vice versa'', the incidence matrix of an incidence structure describes a hypergraph.

The ''incidence matrix'' of an [[incidence structure]] ''C'' is a {{nowrap|''p'' × ''q''}} matrix ''B'' (or its transpose), where ''p'' and ''q'' are the number of ''points'' and ''lines'' respectively, such that {{nowrap|1=''B''_''i'',''j'' = 1}} if the point ''p''_i and line ''L''_''j'' are incident and 0 otherwise. In this case, the incidence matrix is also a [[biadjacency matrix]] of the [[Levi graph]] of the structure. As there is a [[hypergraph]] for every Levi graph, and ''vice versa'', the incidence matrix of an incidence structure describes a hypergraph.

The incidence matrix of an incidence structure C is a  matrix B (or its transpose), where p and q are the number of points and lines respectively, such that  if the point p_i and line L_j are incident and 0 otherwise. In this case, the incidence matrix is also a biadjacency matrix of the Levi graph of the structure. As there is a hypergraph for every Levi graph, and vice versa, the incidence matrix of an incidence structure describes a hypergraph.

The incidence matrix of an incidence structure C is a  matrix B (or its transpose), where p and q are the number of points and lines respectively, such that  if the point p_i and line L_j are incident and 0 otherwise. In this case, the incidence matrix is also a biadjacency matrix of the Levi graph of the structure. As there is a hypergraph for every Levi graph, and vice versa, the incidence matrix of an incidence structure describes a hypergraph.

关联结构 c 的关联矩阵是一个矩阵 b (或其转置) ，其中 p 和 q 分别是点和线的数目，如果点 p < sub > i </sub > 和线 l < sub > j </sub > 是关联的，否则为0。在这种情况下，关联矩阵也是 Levi 图的双邻接矩阵的结构。由于每个 Levi 图都有一个超图，反之亦然，关联结构的关联矩阵描述了一个超图。

关联结构''C''的关联矩阵是一个矩阵''B'' (或其转置) ，其中 p 和 q 分别是点和线的数目，如果点 p_i和线L_j 是关联的，就为1，否则为0。在这种情况下，关联矩阵也是'''<font color="#ff8000">Levi图 Levi Graph</font>'''的双邻接矩阵的结构。由于每个 Levi图都有一个超图，反之亦然。关联结构的关联矩阵描述了一个超图。

===Finite geometries===

==Finite geometries==

'''<font color="#ff8000">有限几何 Finite Geometries</font>'''<br>

An important example is a [[finite geometry]]. For instance, in a finite plane, ''X'' is the set of points and ''Y'' is the set of lines. In a finite geometry of higher dimension, ''X'' could be the set of points and ''Y'' could be the set of subspaces of dimension one less than the dimension of the entire space (hyperplanes); or, more generally, ''X'' could be the set of all subspaces of one dimension ''d'' and ''Y'' the set of all subspaces of another dimension ''e'', with incidence defined as containment.

An important example is a [[finite geometry]]. For instance, in a finite plane, ''X'' is the set of points and ''Y'' is the set of lines. In a finite geometry of higher dimension, ''X'' could be the set of points and ''Y'' could be the set of subspaces of dimension one less than the dimension of the entire space (hyperplanes); or, more generally, ''X'' could be the set of all subspaces of one dimension ''d'' and ''Y'' the set of all subspaces of another dimension ''e'', with incidence defined as containment.

An important example is a finite geometry. For instance, in a finite plane, X is the set of points and Y is the set of lines. In a finite geometry of higher dimension, X could be the set of points and Y could be the set of subspaces of dimension one less than the dimension of the entire space (hyperplanes); or, more generally, X could be the set of all subspaces of one dimension d and Y the set of all subspaces of another dimension e, with incidence defined as containment.

An important example is a finite geometry. For instance, in a finite plane, X is the set of points and Y is the set of lines. In a finite geometry of higher dimension, X could be the set of points and Y could be the set of subspaces of dimension one less than the dimension of the entire space (hyperplanes); or, more generally, X could be the set of all subspaces of one dimension d and Y the set of all subspaces of another dimension e, with incidence defined as containment.

一个重要的例子是有限几何。例如，在有限平面中，x 是点的集合，y 是线的集合。在高维有限几何中，x 可以是点的集合，y 可以是低于整个空间维数的一维子空间(超平面)的集合; 或者，更一般地，x 可以是一维子空间 d 的所有子空间的集合，y 可以是另一维子空间 e 的所有子空间的集合，其关联度定义为包含。

一个有限几何的重要例子。例如，在有限平面中，''X''是点的集合，''Y''是线的集合。在高维有限几何中，''X''可以是点的集合，''Y''可以是低于整个空间维数的一维子空间(超平面)的集合; 或者，更一般地，''X''可以是一维子空间''d''的所有子空间的集合，''Y''可以是另一维子空间''e''的所有子空间的集合，关联度定义也包含在内。

===Polytopes===

==Polytopes==

 

'''<font color="#ff8000">多面体 Polytopes</font>'''

In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.<ref>{{citation|first=H.S.M.|last=Coxeter|author-link=Coxeter|title=[[Regular Polytopes (book)|Regular Polytopes]]|year=1973|edition=3rd|origyear=1963|publisher=Dover|isbn=0-486-61480-8|pages=[https://archive.org/details/regularpolytopes0000coxe/page/166 166-167]}}</ref>

In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.<ref>{{citation|first=H.S.M.|last=Coxeter|author-link=Coxeter|title=[[Regular Polytopes (book)|Regular Polytopes]]|year=1973|edition=3rd|origyear=1963|publisher=Dover|isbn=0-486-61480-8|pages=[https://archive.org/details/regularpolytopes0000coxe/page/166 166-167]}}</ref>

In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.

In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.

 

Another example is a [[block design]]. Here ''X'' is a finite set of "points" and ''Y'' is a class of subsets of ''X'', called "blocks", subject to rules that depend on the type of design. The incidence matrix is an important tool in the theory of block designs. For instance, it can be used to prove [[Fisher's inequality]], a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points.<ref>{{citation|page=99|first=Herbert John|last=Ryser|title=Combinatorial Mathematics|series=The Carus Mathematical Monographs #14|publisher=The Mathematical Association of America|year=1963}}</ref> Considering the blocks as a system of sets, the [[Permanent (mathematics)|permanent]] of the incidence matrix is the number of [[system of distinct representatives|systems of distinct representatives]] (SDRs).

Another example is a [[block design]]. Here ''X'' is a finite set of "points" and ''Y'' is a class of subsets of ''X'', called "blocks", subject to rules that depend on the type of design. The incidence matrix is an important tool in the theory of block designs. For instance, it can be used to prove [[Fisher's inequality]], a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points.<ref>{{citation|page=99|first=Herbert John|last=Ryser|title=Combinatorial Mathematics|series=The Carus Mathematical Monographs #14|publisher=The Mathematical Association of America|year=1963}}</ref> Considering the blocks as a system of sets, the [[Permanent (mathematics)|permanent]] of the incidence matrix is the number of [[system of distinct representatives|systems of distinct representatives]] (SDRs).

Another example is a block design. Here X is a finite set of "points" and Y is a class of subsets of X, called "blocks", subject to rules that depend on the type of design. The incidence matrix is an important tool in the theory of block designs. For instance, it can be used to prove Fisher's inequality, a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points. Considering the blocks as a system of sets, the permanent of the incidence matrix is the number of systems of distinct representatives (SDRs).

Another example is a block design. Here X is a finite set of "points" and Y is a class of subsets of X, called "blocks", subject to rules that depend on the type of design. The incidence matrix is an important tool in the theory of block designs. For instance, it can be used to prove Fisher's inequality, a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points. Considering the blocks as a system of sets, the permanent of the incidence matrix is the number of systems of distinct representatives (SDRs).

另一个例子是块设计。这里 x 是一个有限的“点”集合，而 y 是 x 的一类子集，称为“块” ，受依赖于设计类型的规则的制约。关联矩阵是块设计理论中的一个重要工具。例如，它可以用来证明 Fisher 不等式，一个平衡不完全2- 设计(bibd)的基本定理，块的数目至少是点的数目。将区组看作一个集合系统，关联矩阵的常数是不同代表系统的个数。

另一个例子是块设计。这里''X''是一个有限的“点”集合，而''Y'' 是''x''的一类子集，称为“块” ，受依赖于设计类型的规则的制约。关联矩阵是块设计理论中的一个重要工具。例如，它可以用来证明'''<font color="#ff8000">Fisher不等式 Fisher's inequality</font>'''，一个平衡不完全2- 设计(BIBDs)的基本定理，块的数目至少是点的数目。将块看作一个集合系统，关联矩阵的常数是不同代表系统的个数（SDRs）。

 

 

==References==

==References==