An incidence structure C = ( P , L , I ) {\displaystyle C=(P,L,I)} consists of a set P {\displaystyle P} of points, a set L {\displaystyle L} of lines, and an incidence relation, or set of flags, I ⊆ P × L {\displaystyle I\subseteq P\times L} ; a point p {\displaystyle p} is said to be incident with a line l {\displaystyle l} if ( p , l ) ∈ I {\displaystyle (p,l)\in I} . It is a (finite) partial geometry if there are integers s , t , α ≥ 1 {\displaystyle s,t,\alpha \geq 1} such that:
For any pair of distinct points p {\displaystyle p} and q {\displaystyle q} , there is at most one line incident with both of them. Each line is incident with s + 1 {\displaystyle s+1} points. Each point is incident with t + 1 {\displaystyle t+1} lines. If a point p {\displaystyle p} and a line l {\displaystyle l} are not incident, there are exactly α {\displaystyle \alpha } pairs ( q , m ) ∈ I {\displaystyle (q,m)\in I} , such that p {\displaystyle p} is incident with m {\displaystyle m} and q {\displaystyle q} is incident with l {\displaystyle l} . A partial geometry with these parameters is denoted by p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} .
Properties The number of points is given by ( s + 1 ) ( s t + α ) α {\displaystyle {\frac {(s+1)(st+\alpha )}{\alpha }}} and the number of lines by ( t + 1 ) ( s t + α ) α {\displaystyle {\frac {(t+1)(st+\alpha )}{\alpha }}} . The point graph (also known as the collinearity graph) of a p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} is a strongly regular graph: s r g ( ( s + 1 ) ( s t + α ) α , s ( t + 1 ) , s − 1 + t ( α − 1 ) , α ( t + 1 ) ) {\displaystyle \mathrm {srg} {\Big (}(s+1){\frac {(st+\alpha )}{\alpha }},s(t+1),s-1+t(\alpha -1),\alpha (t+1){\Big )}} . Partial geometries are dualizable structures: the dual of a p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} is simply a p g ( t , s , α ) {\displaystyle \mathrm {pg} (t,s,\alpha )} .
Special cases The generalized quadrangles are exactly those partial geometries p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} with α = 1 {\displaystyle \alpha =1} . The Steiner systems S ( 2 , s + 1 , t s + 1 ) {\displaystyle S(2,s+1,ts+1)} are precisely those partial geometries p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} with α = s + 1 {\displaystyle \alpha =s+1} .
Generalisations A partial linear space S = ( P , L , I ) {\displaystyle S=(P,L,I)} of order s , t {\displaystyle s,t} is called a semipartial geometry if there are integers α ≥ 1 , μ {\displaystyle \alpha \geq 1,\mu } such that:
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