# Partial geometry

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. ).

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 $1 < t < \min ( k - 1 , r - 1 )$.

How to Cite This Entry:
Partial geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_geometry&oldid=51405
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