Partial geometry

An incidence structure C = ( P , L , I ) {displaystyle C=(P,L,I)} consists of points P {displaystyle P} , lines L {displaystyle L} , and flags I ⊆ P × L {displaystyle Isubseteq P imes L} where 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: An incidence structure C = ( P , L , I ) {displaystyle C=(P,L,I)} consists of points P {displaystyle P} , lines L {displaystyle L} , and flags I ⊆ P × L {displaystyle Isubseteq P imes L} where 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: A partial geometry with these parameters is denoted by p g ( s , t , α ) {displaystyle pg(s,t,alpha )} . 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: A semipartial geometry is a partial geometry if and only if μ = α ( t + 1 ) {displaystyle mu =alpha (t+1)} . It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters ( 1 + s ( t + 1 ) + s ( t + 1 ) t ( s − α + 1 ) / μ , s ( t + 1 ) , s − 1 + t ( α − 1 ) , μ ) {displaystyle (1+s(t+1)+s(t+1)t(s-alpha +1)/mu ,s(t+1),s-1+t(alpha -1),mu )} . A nice example of such a geometry is obtained by taking the affine points of P G ( 3 , q 2 ) {displaystyle PG(3,q^{2})} and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters ( s , t , α , μ ) = ( q 2 − 1 , q 2 + q , q , q ( q + 1 ) ) {displaystyle (s,t,alpha ,mu )=(q^{2}-1,q^{2}+q,q,q(q+1))} .