Weakly affine space

In synthetic geometry, weakly affine spaces are a generalization of affine geometries . Emanuel Sperner introduced it at the beginning of the 1960s in order to enable non-Desargue- related levels to be embedded in rooms of higher dimensions . In honor of their inventor, they are also known as Sperner spaces .

definition

A system is given , whereby the elements of are called points , the elements of are called straight lines and the relation between straight lines is called parallelism relation . The set of straight lines should only contain sets of points as elements. This system is called a weakly affine space if the following axioms hold: ${\displaystyle \left({\mathcal {P}},{\mathcal {G}},\parallel \right)}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle \parallel }$ ${\displaystyle {\mathcal {G}}}$

(A1) For two different points there is exactly one straight line that incurs with both points, i.e. for which applies. ${\displaystyle A,B\in {\mathcal {P}}}$ ${\displaystyle g\in {\mathcal {G}}}$ ${\displaystyle A,B\in g}$

(A2) The number of points on each straight line is the same.

(A3) The parallelism relation is an equivalence relation .

(A4) For every pair of a point and a line there is exactly one line with the properties and . ${\displaystyle (P,g)\in {\mathcal {P}}\times {\mathcal {G}}}$ ${\displaystyle h\in {\mathcal {G}}}$ ${\displaystyle h\parallel g}$ ${\displaystyle P\in h}$

Instead of the axiom (A2), a sharper variant is used, which only allows a finite number of points on each straight line and excludes some trivial cases:

(A2e) There is a natural number such that every straight line contains exactly points. ${\displaystyle s\geq 2}$ ${\displaystyle g}$ ${\displaystyle s}$

As with affine planes, the number of points on a straight line is called the order of the weakly affine space.

properties

• In contrast to the axioms of an affine plane (in the axiomatic sense) and an affine space (over a sloping body ), the parallelism in a weakly affine space is not determined in terms of content:

• From (A3) it follows that every straight line is parallel to itself and from the uniqueness of the parallels (A4) it follows that straight lines that are parallel and not identical must actually be disjoint. In general, unlike affine planes and affine block plans , the reverse is not true.

• Every affine plane fulfills the axioms (A1) to (A4) of a weakly affine space, as does every affine geometry.
• Every finite affine plane and every finite affine space satisfies the axioms (A1), (A2e), (A3) and (A4).
• If there is an empty straight line , then (according to A2) it is the only straight line in the weakly affine space. ${\displaystyle g=\emptyset }$
• If there are at least two different lines, then each line contains at least two different points. Therefore a finite weakly affine space is a linear space if and only if it contains more than one straight line. In linear spaces, on the other hand, it is generally not possible to introduce a parallelism relation that satisfies (A3) and (A4).
• Every finite weakly affine space with at least two straight lines fulfills the axioms (A1), (A2e), (A3) and (A4) and is a 2- block diagram ${\displaystyle (v,s,1)}$ , where the straight lines are the blocks and the constant from the axiom is (A2e) . The parallel relation of the weakly affine space is then also a parallelism of the block plan, but the weakly affine space is therefore generally not an affine block plan. ${\displaystyle s}$
• While an affine geometry always satisfies Desargues' theorem if its dimension is greater than 2, this does not have to apply to a weakly affine space. (The definition of a dimension term for weakly affine spaces is somewhat delicate and not very clear. In this statement, dimension should be understood clearly and very limitedly as follows: The dimension of the weakly affine space is less than or equal to 2 if and only if from the disjointness of two straight lines their parallelism always follows.)
• In certain cases it is possible to projectively close even a weakly affine space that is not an affine geometry, analogously to an affine geometry, with a distant space .

Examples

Trivial cases

In the following examples, “space” without additions always means a weakly affine space in the sense of axioms (A1) - (A4). " Affine space " means a space in the sense of linear algebra , that is, an affine space over an inclined body , the usual parallelism applies there, with the projective spaces this relation may have to be added, these are also a projective extension of affine spaces Understand spaces. On the other hand, an “ affine geometry ” is the axiomatically described structure, which only has to be desarguessic if its dimension is greater than 2. An affine plane is synonymous with two-dimensional affine geometry.

• The empty spaces:

• The system forms a space that also fulfills (A2e) empty. It describes an affine and projective space and an affine geometry of dimension −1. ${\displaystyle (\emptyset ,\emptyset ,\emptyset )}$
• Likewise forms a space. This is the only weakly affine space with an empty straight line and neither an affine nor a projective space and also no affine geometry. ${\displaystyle \left(\emptyset ,\{\emptyset \},\{(\emptyset ,\emptyset )\}\right)}$

• The single-point spaces:

• A space consisting of a point without straight lines and thus with an empty parallelism relation is fulfilled (A2e) and is at the same time an affine and projective space and an affine geometry of dimension 0.
• On the other hand, the one point can be a single point on the single straight line in space, the straight line then being parallel to itself. This case does not meet the requirements of an affine or projective space or an affine geometry.

• For any set of points with more than one point there is a space with exactly one straight line which contains all points of the space and is only parallel to itself. All one-dimensional affine and projective spaces have this type and satisfy axioms (A1) to (A4). (In the projective case, the parallelism relation must be supplemented accordingly.) With a finite number of points greater than 1 on the single straight line, the space fulfills the sharper axiom system (A1), (A2e), (A3), (A4) and is always one one-dimensional affine geometry.
• The smallest space with more than one straight line is the minimal model of an affine plane . This is a linear and affine space, but not a projective space.

A class of finite models

Be a finite Links Almost body , . Then you use it as a point set . A straight line is determined by a point and the multiple of a "vector" , that is, by the parameter equation ${\displaystyle K}$ ${\displaystyle n\in \mathbb {N} ,n\geq 2}$ ${\displaystyle K^{n}}$ ${\displaystyle g}$ ${\displaystyle p_{0}\in K^{n}}$ ${\displaystyle v\in K^{n}\setminus \lbrace 0\rbrace }$

${\displaystyle g:x=p_{0}+v\cdot \lambda :\quad \lambda \in K.}$

Two straight lines with the generating vectors are parallel if and only if there are scalars with which applies. ${\displaystyle v,w\in K^{n}\setminus \lbrace 0\rbrace }$ ${\displaystyle \lambda ,\mu \in K\setminus \lbrace 0\rbrace }$ ${\displaystyle v\cdot \lambda +w\cdot \mu =0}$

The spaces defined in this way are weakly affine and satisfy the more stringent system of axioms (A1), (A2e), (A3), (A4).

• For these spaces are affine translation planes and desarguessic if and only if a body is. ${\displaystyle n=2}$ ${\displaystyle K}$
• For they are only affine spaces (and then of course also affine geometries) if is a body. ${\displaystyle n>2}$ ${\displaystyle K}$

Infinite example

Let the set of points , the grid of points with integer coordinates in the real plane, be the set of rational lines, i.e. the set of lines in the affine plane over the rational numbers . Every rational straight line that contains a point with integer coordinates meets an infinite number of such points, therefore the set of lines fulfills the axiom (A2), every “straight line” from is countably infinite . The parallelism relation should be the one inherited from the rational “carrier line” . Then there is an infinite weakly affine space. It is neither an affine nor a projective space and neither is it an affine geometry. ${\displaystyle {\mathcal {P}}=\mathbb {Z} ^{2}}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \mathbb {Q} ^{2}}$ ${\displaystyle {\mathcal {G}}=\lbrace h\cap \mathbb {Z} ^{2}:\;h\in {\mathcal {H}}\rbrace \setminus \lbrace \emptyset \rbrace }$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle \parallel }$ ${\displaystyle \left(g_{1}\parallel g_{2}\Leftrightarrow \exists h_{1},h_{2}\in {\mathcal {H}}:g_{1}\subset h_{1}\land g_{2}\subset h_{2}\land h_{1}\parallel h_{2})\right)}$ ${\displaystyle \left({\mathcal {P}},{\mathcal {G}},\parallel \right)}$

literature

• Hans-Joachim Arnold : About distant spaces with weakly affine spaces . In: Treatises from the Mathematical Seminar of the University of Hamburg . tape 30 , no. 1-2 . University of Berlin, Hamburg 1967, p. 75-105 , doi : 10.1007 / BF02993993 .
• Tomáš Kepka, Petr Němec: Trilinear constructions of quasimodules . In: Commentationes Mathematicae Universitatis Carolinae . Vol. 21, No. 2 , 1980, ISSN  1213-7243 , pp. 341-354 ( online ).
• Aleksandar Samardžiski: A class of finite Sperner spaces . In: Treatises from the Mathematical Seminar of the University of Hamburg . Volume 42, No. 1 , ISSN  1435-5345 , p. 205-211 , doi : 10.1007 / BF02993547 .
• Werner Seier: A characterization of the weakly affine spaces using scalar systems . In: manuscripta mathematica . Volume 8, No. 1 , 1960, ISSN  1435-5345 , pp. 39-57 , doi : 10.1007 / BF01317576 .
• Emanuel Sperner : Affine Spaces with Weak Incidence and Associated Algebraic Structures . In: Journal for Pure and Applied Mathematics (Crelles Journal) . tape 1960 , no. 204 . University of Berlin, 1960, ISSN  1435-5345 , p. 205-215 , doi : 10.1515 / crll.1960.204.205 .
• Emanuel Sperner: On non-desarguesian geometries . In: Seminari dell'Istituto Nazionale di Alta Matematica . Series 3, volume 17 , no. 4 . UNIONE MATEMATICA ITALIANA, Roma 1962 ( online [PDF; accessed June 25, 2012]).

Individual evidence

1. ↑ Sperner (1960) and (1964)
2. ↑ ^{a b} Samardžiski (1972)
3. ↑ Arnold (1967)
4. ↑ is referred to by Sperner (1960) as a “generalized vector space”, by Kepka and Němec (1980) as a quasi-module above . ${\displaystyle K^{n}}$ ${\displaystyle K}$