Linear space (geometry)
A linear space , sometimes referred to as an incidence space, is a fundamental structure in finite geometry . It was introduced as an independent term in 1964 by Paul Libois . Except in trivial cases (with at most one-dimensional spaces) one can view linear spaces as a generalization of the weakly affine spaces , which in turn are a generalization of the affine spaces . At the same time, linear spaces also represent a generalization of the at least two-dimensional projective spaces . Finite linear spaces, in turn, can be viewed as a generalization of 2- block plans , in which one waives the fact that the same number of points lie on each straight line (= block).
Definitions
Linear space
Let be an incidence structure where the elements of are called points and the elements of are called straight lines (or blocks ). In addition, one uses for the incidence relation also the ways of speaking a point lies on a straight line ( with ) and a straight line goes through a point ( with ). The incidence structure is called linear space if the following 3 axioms are met:
- (L1) Exactly one straight line goes through 2 points.
- (L2) There are at least 2 points on every straight line.
- (L3) L has at least 2 straight lines.
Occasionally, the axiom L3 is not required in the literature, in such a case those linear spaces that still satisfy it are called non-trivial linear spaces. If the linear space has a finite number of points, one also speaks briefly of a finite linear space.
Almost always only finite linear spaces are examined. As with block plans , the number of points is usually denoted by and the number of straight lines by .
Partial linear space
For an incidence structure, axioms (L2) and (L3) apply, but instead of (L1) only the weaker axiom
- (L1p) "At most one straight line goes through 2 points .",
so one speaks of a partial linear space (eng. partial linear space ).
Degenerate linear space, near-pencil
- A linear space with points that has a straight line with points is called degenerate linear space ( degenerate linear space or near-pencil ). All other linear spaces are called nondegenerate linear spaces .
- The class of the projective planes is combined with the class of the degenerate linear spaces to form the class of the generalized projective plane .
properties
- Every linear space is a simple incidence structure, that is, a straight line is uniquely determined by the points which it incurs, so it can be understood as a set of its points.
- For finite linear spaces the number of lines is never less than the number of points, so it always applies . That is the statement of a sentence by de Bruijn and Erdős .
- Equality applies if and only if the linear space is a generalized projective plane, i.e. either a projective plane or a near-pencil.
- The maximum possible number of straight lines for a given number of points is . The linear space is then the complete graph on v nodes.
Examples
- The normal Euclidean plane forms an infinite linear space.
- Somewhat more general are all affine and projective spaces whose dimensions are greater than or equal to 2, and thus in particular also projective levels, such as the Fano level , (non-trivial) linear spaces.
- A dotted projective plane is created from a projective plane by omitting exactly one point: Such a plane always forms a linear space.
- An affine plane with a far point is created from an affine plane by adding exactly one far point as the intersection point of exactly a fixed set of parallels of the plane. These levels also always form a linear space.
All four linear spaces are listed below with five dots ( ). Here it is common not to draw all straight lines with only two points in the graphic representation for reasons of clarity.
10 lines: Complete graph on 5 nodes . | 8 straights | 6 lines: the affine plane with a far point. | 5 straight lines: the near pencil with 5 points. |
near pencil with 10 points (10 lines) |
literature
- Albrecht Beutelspacher : Introduction to Finite Geometry . tape II . Bibliographisches Institut, 1983, ISBN 3-411-01648-5 , pp. 159 .
- Klaus Metsch: Linear Spaces with Few Lines . Springer, Berlin / Heidelberg / New York / London / Paris / Tokyo / Hong Kong / Barcelona / Budapest 1991, ISBN 3-540-54720-7 .
- Jacobus Hendricus van Lint , Richard M. Wilson : A Course in Combinatorics . Cambridge University Press, 1992, ISBN 0-521-42260-4 , pp. 188 .
- LM Batten, Albrecht Beutelspacher: The Theory of Finite Linear Spaces . Cambridge University Press, Cambridge 1992, ISBN 978-0-521-11418-9 .
Web links
- Geometry script on math.uni-hamburg.de (PDF file; 326 kB)
- Script on incidence geometry ( memento from February 6, 2012 in the Internet Archive ) on mathematik.uni-tuebingen.de