Partial linear space

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        S
        =
        (
        
          
            P
          
        
        ,
        
          
            L
          
        
        ,
        
          
            I
          
        
        )
      
    
    {\displaystyle S=({\mathcal {P}},{\mathcal {L}},{\textbf {I}})}
  
 an incidence structure, for which the elements of 
  
    
      
        
          
            P
          
        
      
    
    {\displaystyle {\mathcal {P}}}
  
 are called <i>points</i> and the elements of 
  
    
      
        
          
            L
          
        
      
    
    {\displaystyle {\mathcal {L}}}
  
 are called <i>lines</i>. <i>S</i> is a partial linear space, if the following axioms hold:
</p>
<ul><li>any line is incident with at least two points</li>
<li>any pair of distinct points is incident with at most one line</li></ul>
<p>If there is a unique line incident with every pair of distinct points, then we get a linear space. 
</p>
<h2 id="properties">Properties</h2>
<p>The <a href="/facts/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(incidence_geometry)/bHJzYWjL">De Bruijn–Erdős theorem</a> shows that in any finite linear space 
  
    
      
        S
        =
        (
        
          
            P
          
        
        ,
        
          
            L
          
        
        ,
        
          
            I
          
        
        )
      
    
    {\displaystyle S=({\mathcal {P}},{\mathcal {L}},{\textbf {I}})}
  
 which is not a single point or a single line, we have 
  
    
      
        
          |
        
        
          
            P
          
        
        
          |
        
        ≤
        
          |
        
        
          
            L
          
        
        
          |
        
      
    
    {\displaystyle |{\mathcal {P}}|\leq |{\mathcal {L}}|}
  
. 
</p>
<h2 id="examples">Examples</h2>
<ul><li><a href="/facts/Projective_space/7MSpA9cM">Projective space</a></li>
<li><a href="/facts/Affine_space/tlvI3oX1">Affine space</a></li>
<li><a href="/facts/Polar_space/EF8Dyqqp">Polar space</a></li>
<li><a href="/facts/Generalized_quadrangle/nFRoYCIc">Generalized quadrangle</a></li>
<li><a href="/facts/Generalized_polygon/ehKVCuyl">Generalized polygon</a></li>
<li><a href="/facts/Near_polygon/wb3a5FVq">Near polygon</a></li></ul>

<ul><li>Shult, Ernest E. (2011), <i>Points and Lines</i>, Universitext, Springer, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-15627-4">10.1007/978-3-642-15627-4</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-15626-7.</li></ul>
<ul><li><a href="/facts/Lynn_Batten/Gew5MGW3">Lynn Batten</a>: <i><a href="/facts/Combinatorics_of_Finite_Geometries/r5zgGsmd">Combinatorics of Finite Geometries</a></i>. Cambridge University Press 1986, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-31857-2, p. 1-22</li>
<li><a href="/facts/Lynn_Batten/Gew5MGW3">Lynn Batten</a> and  <a href="/facts/Albrecht_Beutelspacher/I0l9b5so">Albrecht Beutelspacher</a>: The Theory of Finite Linear Spaces. Cambridge University Press, Cambridge, 1992.</li>
<li>Eric Moorhouse: <a href="https://web.archive.org/web/20131029221809/http://www.uwyo.edu/moorhouse/handouts/incidence_geometry.pdf"><i>Incidence Geometry</i></a>. Lecture notes (archived)</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.math.uni-kiel.de/geometrie/klein/math/geometry/partlin.html"><i>partial linear space</i></a> at the University of Kiel</li>
<li><a href="https://planetmath.org/encyclopedia/PartialLinearSpace.html"><i>partial linear space</i></a> at <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a></li></ul>

Partial linear space open-in-new

Partial linear space