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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 Point (geometry), points and Line (geometry), lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the heterogeneous relation, relation of which points are incident (geometry), 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 (such as affine plane (incidence geometry), affine, projective plane, projective, and Möbius 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 (Plane (mathematics), planes, Solid geometry, sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Plane
In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements. The standard notation for this plane, as a member of a family of projective spaces, is . Here, stands for "projective geometry", the first parameter is the geometric dimension (it is a plane, of dimension 2) and the second parameter is the order (the number of points per line, minus one). The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study. In a separate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (projective Geometry)
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by Point (geometry), points and Line (geometry), lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special Map (mathematics), mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a ''duality''. Such a map can be constructed in many ways. The concept of plane duality readily extends to ''space duality'' and beyond that to duality in any finite-dimensional projective geometry. Principle of duality A projective plane may be defined axiomatically as an incidence structure, in terms of a set of ''points'', a set of ''lines'', and an incidence relation that de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse Relation
In mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if X and Y are sets and L \subseteq X \times Y is a relation from X to Y, then L^ is the relation defined so that yL^x if and only if xLy. In set-builder notation, :L^ = \. Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation is also called the transpose relation. It has also been called the opposite or dual of the original relation, the inverse of the original relation,Gerard O'Regan (2016): ''Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications'' or the reciprocal L^ of the relation L. Other notations for the converse relation include L^, L^, \breve, L^, or L^. The notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Plane
In geometry, an affine plane is a two-dimensional affine space. Definitions There are two ways to formally define affine planes, which are equivalent for affine planes over a field. The first way consists in defining an affine plane as a set on which a vector space of dimension two acts simply transitively. Intuitively, this means that an affine plane is a vector space of dimension two in which one has "forgotten" where the origin is. The second way occurs in incidence geometry, where an affine plane is defined as an abstract system of points and lines satisfying a system of axioms. Coordinates and isomorphism All the affine planes defined over a field are isomorphic. More precisely, the choice of an affine coordinate system (or, in the real case, a Cartesian coordinate system) for an affine plane P over a field F induces an isomorphism of affine planes between P and F^2. In the more general situation, where the affine planes are not defined over a field, they will in gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Space (geometry)
A linear space is a basic structure in incidence geometry. A linear space consists of a set of elements called points, and a set of elements called lines. Each line is a distinct subset of the points. The points in a line are said to be incident with the line. Each two points are in a line, and any two lines may have no more than one point in common. Intuitively, this rule can be visualized as the property that two straight lines never intersect more than once. Linear spaces can be seen as a generalization of projective and affine planes, and more broadly, of 2-(v,k,1) block designs, where the requirement that every block contains the same number of points is dropped and the essential structural characteristic is that 2 points are incident with exactly 1 line. The term ''linear space'' was coined by Paul Libois in 1964, though many results about linear spaces are much older. Definition Let ''L'' = (''P'', ''G'', ''I'') be an incidence structure, for which the elements of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Linear Space
A partial linear space (also semilinear or near-linear space) is a basic incidence structure in the field of incidence geometry, that carries slightly less structure than a linear space. The notion is equivalent to that of a linear hypergraph. Definition Let S=(,, \textbf) an incidence structure, for which the elements of are called ''points'' and the elements of are called ''lines''. ''S'' is a partial linear space, if the following axioms hold: * any line is incident with at least two points * any pair of distinct points is incident with at most one line If there is a unique line incident with every pair of distinct points, then we get a linear space. Properties The De Bruijn–Erdős theorem shows that in any finite linear space S=(,, \textbf) which is not a single point or a single line, we have , \mathcal, \leq , \mathcal, . Examples * Projective space * Affine space * Polar space * Generalized quadrangle * Generalized polygon * Near polygon References * . * Lyn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Edges
In graph theory, multiple edges (also called parallel edges or a multi-edge), are, in an undirected graph, two or more edges that are incident to the same two vertices, or in a directed graph, two or more edges with both the same tail vertex and the same head vertex. A simple graph has no multiple edges and no loops. Depending on the context, a graph may be defined so as to either allow or disallow the presence of multiple edges (often in concert with allowing or disallowing loops): *Where graphs are defined so as to ''allow'' multiple edges and loops, a graph without loops or multiple edges is often distinguished from other graphs by calling it a ''simple graph.'' *Where graphs are defined so as to ''disallow'' multiple edges and loops, a multigraph or a pseudograph is often defined to mean a "graph" which ''can'' have multiple edges. Multiple edges are, for example, useful in the consideration of electrical networks, from a graph theoretical point of view. Additional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop (graph Theory)
In graph theory, a loop (also called a self-loop or a ''buckle'') is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple edges between the same vertices): * Where graphs are defined so as to ''allow'' loops and multiple edges, a graph without loops or multiple edges is often distinguished from other graphs by calling it a ''simple graph''. * Where graphs are defined so as to ''disallow'' loops and multiple edges, a graph that does have loops or multiple edges is often distinguished from the graphs that satisfy these constraints by calling it a ''multigraph'' or ''pseudograph''. In a graph with one vertex, all edges must be loops. Such a graph is called a bouquet. Degree For an undirected graph, the degree of a vertex is equal to the number of adjacent vertices. A special ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Graph
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
