1#an example of an incidence structure that is not an incidence geometry 2pg := PG(4,2); 3pg2 := PG(9,2); 4points := List(Lines(pg),x->VectorSpaceToElement(pg2,GrassmannCoordinates(x)));; 5flags := Concatenation(List(Points(pg),x->List(Planes(x),y->FlagOfIncidenceStructure(pg,[x,y]))));; 6prelines := List(flags,flag->ShadowOfFlag(pg,flag,2));; 7lines := List(prelines,x->VectorSpaceToElement(pg2,List(x,y->GrassmannCoordinates(y))));; 8flags2 := Concatenation(List(Points(pg),x->List(Solids(x),y->FlagOfIncidenceStructure(pg,[x,y]))));; 9preplanes := List(flags2,flag->ShadowOfFlag(pg,flag,2));; 10planes := List(preplanes,x->VectorSpaceToElement(pg2,List(x,y->GrassmannCoordinates(y))));; 11maximals1 := List(Planes(pg),x->VectorSpaceToElement(pg2,List(Lines(x),y->GrassmannCoordinates(y))));; 12maximals2 := List(Points(pg),x->VectorSpaceToElement(pg2,List(Lines(x),y->GrassmannCoordinates(y))));; 13elements := Union(points,lines,planes,maximals1,maximals2);; 14Length(elements); 15type := x -> ProjectiveDimension(x)+1; 16inc_rel := \*; 17inc := IncidenceStructure(elements,inc_rel,type,[1,2,3,4]); 18Rank(inc); 19quit; 20