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An incidence structure (cf. [[Incidence system|Incidence system]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p0716501.png" /> in which the incidence relation between points and lines is symmetric and satisfies the following axioms:
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1) each point is incident to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p0716502.png" /> lines, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p0716503.png" />, and two distinct points are incident to at most one line;
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{{TEX|done}}
  
2) each line is incident to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p0716504.png" /> points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p0716505.png" />;
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An incidence structure (cf. [[Incidence system|Incidence system]]) $  S = ( P , L , I ) $
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in which the incidence relation between points and lines is symmetric and satisfies the following axioms:
  
3) through each point not incident to a line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p0716506.png" /> there are exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p0716507.png" /> lines intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p0716508.png" />.
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1) each point is incident to $  r $
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lines,  $  r \geq  2 $,
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and two distinct points are incident to at most one line;
  
If a partial geometry consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p0716509.png" /> points and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165010.png" /> lines, then
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2) each line is incident to  $  k $
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points, $  k \geq  2 $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165011.png" /></td> </tr></table>
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3) through each point not incident to a line  $  l $
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there are exactly  $  t \geq  1 $
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lines intersecting  $  l $.
  
and necessary conditions for the existence of such a partial geometry are that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165012.png" /> be divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165014.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165016.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165017.png" /> (cf. [[#References|[2]]]).
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If a partial geometry consists of  $  v $
 +
points and  $  b $
 +
lines, then
 +
 
 +
$$
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v  = 
 +
\frac{k [ ( k - 1 ) ( r - 1 ) + t ] }{t}
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\  \textrm{ and } \ \
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b  = 
 +
\frac{r [ ( k - 1 ) ( r - 1 ) + t ] }{t}
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,
 +
$$
 +
 
 +
and necessary conditions for the existence of such a partial geometry are that $  ( k - 1 ) ( r - 1 ) k r $
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be divisible by $  t ( k + r - t - 1 ) $,  
 +
$  k ( k - 1 ) ( r - 1 ) $
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by $  t $
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and $  r ( k - 1 ) ( r - 1 ) $
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by $  t $(
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cf. [[#References|[2]]]).
  
 
Partial geometries can be divided into four classes:
 
Partial geometries can be divided into four classes:
  
a) partial geometries with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165018.png" /> or (dually) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165019.png" />. Geometries of this type are just <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165020.png" />-schemes or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165021.png" />-schemes (cf. [[Block design|Block design]]);
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a) partial geometries with $  t = k $
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or (dually) $  t = r $.  
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Geometries of this type are just $  2 - ( v , k , 1 ) $-
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schemes or $  2 - ( v , r , 1 ) $-
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schemes (cf. [[Block design|Block design]]);
  
b) partial geometries with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165022.png" /> or (dually) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165023.png" />. In this case a partial geometry is the same thing as a net of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165024.png" /> and defect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165025.png" /> (or dually);
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b) partial geometries with $  t = k - 1 $
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or (dually) $  t = r - 1 $.  
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In this case a partial geometry is the same thing as a net of order $  k $
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and defect $  k - r + 1 $(
 +
or dually);
  
c) partial geometries with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165026.png" />, known as generalized quadrangles;
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c) partial geometries with $  t = 1 $,  
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known as generalized quadrangles;
  
d) partial geometries with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165027.png" />.
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d) partial geometries with $  1 < t < \min ( k - 1 , r - 1 ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.C. Bose,  "Strongly regular graphs, partial geometries and partially balanced designs"  ''Pacific J. Math.'' , '''13''' :  2  (1963)  pp. 389–419</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Thas,  "Combinatorics of partial geometries and generalized quadrangles"  M. Aigner (ed.) , ''Higher Combinatorics'' , Reidel  (1977)  pp. 183–199</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Thas,  "Construction of maximal arcs and partial geometries"  ''Geometrica Dedicata'' , '''3''' :  1  (1974)  pp. 61–64</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.C. Bose,  "Strongly regular graphs, partial geometries and partially balanced designs"  ''Pacific J. Math.'' , '''13''' :  2  (1963)  pp. 389–419</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Thas,  "Combinatorics of partial geometries and generalized quadrangles"  M. Aigner (ed.) , ''Higher Combinatorics'' , Reidel  (1977)  pp. 183–199</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Thas,  "Construction of maximal arcs and partial geometries"  ''Geometrica Dedicata'' , '''3''' :  1  (1974)  pp. 61–64</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
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At present there are only two infinite series and a few sporadic examples of partial geometries of type d) known. One of these series is related to maximal arcs in projective planes (cf. [[#References|[3]]]) and the other to hyperbolic quadrics in projective spaces of characteristic 2 (cf. [[#References|[a1]]]).
 
At present there are only two infinite series and a few sporadic examples of partial geometries of type d) known. One of these series is related to maximal arcs in projective planes (cf. [[#References|[3]]]) and the other to hyperbolic quadrics in projective spaces of characteristic 2 (cf. [[#References|[a1]]]).
  
There is an important connection to strongly-regular graphs: The point graph (which has the points of the partial geometry as vertices, with two points being adjacent if and only if they are collinear in the partial geometry) is a strongly-regular graph, cf. [[#References|[1]]]. This allows one to apply the many known existence criteria for such graphs to partial geometries. A good survey on strongly-regular graphs and partial geometries, containing non-existence results and descriptions of the known examples, has been given in [[#References|[a2]]]. For the special case of generalized quadrangles, there is now a monograph available, see [[#References|[a3]]].
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There is an important connection to [[strongly regular graph]]s: The point graph (which has the points of the partial geometry as vertices, with two points being adjacent if and only if they are collinear in the partial geometry) is a strongly-regular graph, cf. [[#References|[1]]]. This allows one to apply the many known existence criteria for such graphs to partial geometries. A good survey on strongly-regular graphs and partial geometries, containing non-existence results and descriptions of the known examples, has been given in [[#References|[a2]]]. For the special case of generalized quadrangles, there is now a monograph available, see [[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. de Clerck,  R.H. Dye,  J.A. Thas,  "An infinite class of partial geometries associated with the hyperbolic quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165028.png" />"  ''Europ. J. Comb.'' , '''1'''  (1980)  pp. 323–326</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.E. Brouwer,  J.H. van Lint,  "Strongly regular graphs and partial geometries"  D.M. Jackson (ed.)  S.A. Vanstone (ed.) , ''Enumeration and Design'' , Acad. Press  (1984)  pp. 85–122</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.E. Payne,  J.A. Thas,  "Finite generalized quadrangles" , Pitman  (1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.E. Brouwer,  A.M. Cohen,  A. Neumaier,  "Distance regular graphs" , Springer  (1989)  pp. 229</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.M. Batten,  "Combinatorics of finite geometries" , Cambridge Univ. Press  (1986)  pp. Chapt. 7</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.H. van Lint,  "Partial geometries" , ''Proc. Internat. Congress Mathematicians (Warsawa 1983)'' , '''2''' , PWN &amp; North-Holland  (1984)  pp. 1579–1590</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. de Clerck,  R.H. Dye,  J.A. Thas,  "An infinite class of partial geometries associated with the hyperbolic quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071650/p07165028.png" />"  ''Europ. J. Comb.'' , '''1'''  (1980)  pp. 323–326</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.E. Brouwer,  J.H. van Lint,  "Strongly regular graphs and partial geometries"  D.M. Jackson (ed.)  S.A. Vanstone (ed.) , ''Enumeration and Design'' , Acad. Press  (1984)  pp. 85–122</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.E. Payne,  J.A. Thas,  "Finite generalized quadrangles" , Pitman  (1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.E. Brouwer,  A.M. Cohen,  A. Neumaier,  "Distance regular graphs" , Springer  (1989)  pp. 229</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.M. Batten,  "Combinatorics of finite geometries" , Cambridge Univ. Press  (1986)  pp. Chapt. 7</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.H. van Lint,  "Partial geometries" , ''Proc. Internat. Congress Mathematicians (Warsawa 1983)'' , '''2''' , PWN &amp; North-Holland  (1984)  pp. 1579–1590</TD></TR></table>

Latest revision as of 09:31, 19 January 2021


An incidence structure (cf. Incidence system) $ S = ( P , L , I ) $ in which the incidence relation between points and lines is symmetric and satisfies the following axioms:

1) each point is incident to $ r $ lines, $ r \geq 2 $, and two distinct points are incident to at most one line;

2) each line is incident to $ k $ points, $ k \geq 2 $;

3) through each point not incident to a line $ l $ there are exactly $ t \geq 1 $ lines intersecting $ l $.

If a partial geometry consists of $ v $ points and $ b $ lines, then

$$ v = \frac{k [ ( k - 1 ) ( r - 1 ) + t ] }{t} \ \textrm{ and } \ \ b = \frac{r [ ( k - 1 ) ( r - 1 ) + t ] }{t} , $$

and necessary conditions for the existence of such a partial geometry are that $ ( k - 1 ) ( r - 1 ) k r $ be divisible by $ t ( k + r - t - 1 ) $, $ k ( k - 1 ) ( r - 1 ) $ by $ t $ and $ r ( k - 1 ) ( r - 1 ) $ by $ t $( cf. [2]).

Partial geometries can be divided into four classes:

a) partial geometries with $ t = k $ or (dually) $ t = r $. Geometries of this type are just $ 2 - ( v , k , 1 ) $- schemes or $ 2 - ( v , r , 1 ) $- schemes (cf. Block design);

b) partial geometries with $ t = k - 1 $ or (dually) $ t = r - 1 $. In this case a partial geometry is the same thing as a net of order $ k $ and defect $ k - r + 1 $( or dually);

c) partial geometries with $ t = 1 $, known as generalized quadrangles;

d) partial geometries with $ 1 < t < \min ( k - 1 , r - 1 ) $.

References

[1] R.C. Bose, "Strongly regular graphs, partial geometries and partially balanced designs" Pacific J. Math. , 13 : 2 (1963) pp. 389–419
[2] J.A. Thas, "Combinatorics of partial geometries and generalized quadrangles" M. Aigner (ed.) , Higher Combinatorics , Reidel (1977) pp. 183–199
[3] J.A. Thas, "Construction of maximal arcs and partial geometries" Geometrica Dedicata , 3 : 1 (1974) pp. 61–64

Comments

For nets see also Net (in finite geometry).

At present there are only two infinite series and a few sporadic examples of partial geometries of type d) known. One of these series is related to maximal arcs in projective planes (cf. [3]) and the other to hyperbolic quadrics in projective spaces of characteristic 2 (cf. [a1]).

There is an important connection to strongly regular graphs: The point graph (which has the points of the partial geometry as vertices, with two points being adjacent if and only if they are collinear in the partial geometry) is a strongly-regular graph, cf. [1]. This allows one to apply the many known existence criteria for such graphs to partial geometries. A good survey on strongly-regular graphs and partial geometries, containing non-existence results and descriptions of the known examples, has been given in [a2]. For the special case of generalized quadrangles, there is now a monograph available, see [a3].

References

[a1] F. de Clerck, R.H. Dye, J.A. Thas, "An infinite class of partial geometries associated with the hyperbolic quadric in " Europ. J. Comb. , 1 (1980) pp. 323–326
[a2] A.E. Brouwer, J.H. van Lint, "Strongly regular graphs and partial geometries" D.M. Jackson (ed.) S.A. Vanstone (ed.) , Enumeration and Design , Acad. Press (1984) pp. 85–122
[a3] S.E. Payne, J.A. Thas, "Finite generalized quadrangles" , Pitman (1985)
[a4] A.E. Brouwer, A.M. Cohen, A. Neumaier, "Distance regular graphs" , Springer (1989) pp. 229
[a5] L.M. Batten, "Combinatorics of finite geometries" , Cambridge Univ. Press (1986) pp. Chapt. 7
[a6] J.H. van Lint, "Partial geometries" , Proc. Internat. Congress Mathematicians (Warsawa 1983) , 2 , PWN & North-Holland (1984) pp. 1579–1590
How to Cite This Entry:
Partial geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_geometry&oldid=11436
This article was adapted from an original article by V.V. Afanas'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article