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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 * . *Lynn Bat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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, 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 (planes, solids, -spaces, conics, etc.) can be used. The study of finite structures is sometimes called finite geometry. Formal definition and terminology An incidence structure is a triple ( ... [...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. 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 ''P'' are called points and the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergraph
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) where X is a set of elements called ''nodes'' or ''vertices'', and E is a set of non-empty subsets of X called ''hyperedges'' or ''edges''. Therefore, E is a subset of \mathcal(X) \setminus\, where \mathcal(X) is the power set of X. The size of the vertex set is called the ''order of the hypergraph'', and the size of edges set is the ''size of the hypergraph''. A directed hypergraph differs in that its hyperedges are not sets, but ordered pairs of subsets of X, with each pair's first and second entries constituting the tail and head of the hyperedge respectively. While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same card ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Bruijn–Erdős Theorem (incidence Geometry)
In incidence geometry, the De Bruijn–Erdős theorem, originally published by , states a lower bound on the number of lines determined by ''n'' points in a projective plane. By Duality (projective geometry), duality, this is also a bound on the number of intersection points determined by a configuration of lines. Although the proof given by De Bruijn and Erdős is Combinatorial proof, combinatorial, De Bruijn and Erdős noted in their paper that the analogous (Euclidean geometry, Euclidean) result is a consequence of the Sylvester–Gallai theorem, by an Mathematical induction, induction on the number of points. Statement of the theorem Let ''P'' be a configuration of ''n'' points in a projective plane, not all on a line. Let ''t'' be the number of lines determined by ''P''. Then, * ''t'' ≥ ''n'', and * if ''t'' = ''n'', any two lines have exactly one point of ''P'' in common. In this case, ''P'' is either a projective plane or ''P'' is a ''near pencil'', meaning that e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead ''displacement vectors'', also called ''translation'' vectors or simply ''translations'', between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be viewed as an affine spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Space
In mathematics, in the field of geometry, a polar space of rank ''n'' (), or ''projective index'' , consists of a set ''P'', conventionally called the set of points, together with certain subsets of ''P'', called ''subspaces'', that satisfy these axioms: * Every subspace is isomorphic to a projective geometry with and ''K'' a division ring. By definition, for each subspace the corresponding ''d'' is its dimension. * The intersection of two subspaces is always a subspace. * For each point ''p'' not in a subspace ''A'' of dimension of , there is a unique subspace ''B'' of dimension containing ''p'' and such that is -dimensional. The points in are exactly the points of ''A'' that are in a common subspace of dimension 1 with ''p''. * There are at least two disjoint subspaces of dimension . It is possible to define and study a slightly bigger class of objects using only relationship between points and lines: a polar space is a partial linear space (''P'',''L''), so that for each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Quadrangle
In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are the with ''n'' = 4 and near 2n-gons with ''n'' = 2. They are also precisely the partial geometries pg(''s'',''t'',α) with α = 1. Definition A generalized quadrangle is an incidence structure (''P'',''B'',I), with I ⊆ ''P'' × ''B'' an incidence relation, satisfying certain axioms. Elements of ''P'' are by definition the ''points'' of the generalized quadrangle, elements of ''B'' the ''lines''. The axioms are the following: * There is an ''s'' (''s'' ≥ 1) such that on every line there are exactly ''s'' + 1 points. There is at most one point on two distinct lines. * There is a ''t'' (''t'' ≥ 1) such that through every point there are exactly ''t'' + 1 lines. There is at most one line through two distinct points. * For every point ''p'' not on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Polygon
In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized ''n''-gons encompass as special cases projective planes (generalized triangles, ''n'' = 3) and generalized quadrangles (''n'' = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the '' Moufang property'' have been completely classified by Tits and Weiss. Every generalized ''n''-gon with ''n'' even is also a near polygon. Definition A generalized ''2''-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line. For ''n \geq 3'' a generalized ''n''-gon is an incidence structure (P,L,I), where P is the set of points, L is the set of lines and I\subseteq P\times L is the incidence relation, such that: * It is a partial linear space. * It has no ordinary ''m''-gons as subge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Near Polygon
In mathematics, a near polygon is an incidence geometry introduced by Ernest E. Shult and Arthur Yanushka in 1980. Shult and Yanushka showed the connection between the so-called tetrahedrally closed line-systems in Euclidean spaces and a class of point-line geometries which they called near polygons. These structures generalise the notion of generalized polygon as every generalized 2''n''-gon is a near 2''n''-gon of a particular kind. Near polygons were extensively studied and connection between them and dual polar spaces was shown in 1980s and early 1990s. Some sporadic simple groups, for example the Hall-Janko group and the Mathieu groups, act as automorphism groups of near polygons. Definition A near 2''d''-gon is an incidence structure (P,L,I), where P is the set of points, L is the set of lines and I\subseteq P\times L is the incidence relation, such that: * The maximum distance between two points (the so-called diameter) is ''d''. * For every point x and every line L the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lynn Batten
Lynn Margaret Batten (1948 – 28 July 2022 The Age, 2 August 2022 (death notice)) was a Canadian-Australian mathematician known for her books about and , and for her research on the classification of malware. Education and career Batten earned her Ph.D. at the |
Combinatorics Of Finite Geometries
''Combinatorics of Finite Geometries'' is an undergraduate mathematics textbook on finite geometry by Lynn Batten. It was published by Cambridge University Press in 1986 with a second edition in 1997 (). Topics The types of finite geometry covered by the book include partial linear spaces, linear spaces, affine spaces and affine planes, projective spaces and projective planes, polar spaces, generalized quadrangles, and partial geometries. A central connecting concept is the "connection number" of a point and a line not containing it, equal to the number of lines that meet the given point and intersect the given line. The second edition adds a final chapter on blocking sets. Beyond the basic theorems and proofs of this subject, the book includes many examples and exercises, and some history and information about current research. Audience and reception The book is aimed at advanced undergraduates, assuming only an introductory-level of abstract algebra and some knowledge of line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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