Generalized Fischer spaces

Abstract

Abstract A generalized Fischer space is a partial linear space in which any two intersecting lines generate a subspace isomorphic to an affine plane or the dual of an affine plane. We give a classification of all finite and infinite generalized Fischer spaces under some nondegeneracy conditions. Introduction Let Π = ( P, L ) be a partial linear space , that is a set of points P together with a set L of subsets of P of cardinality at least 2 called lines , such that each pair of points is in at most one line. A subset X of P is called a subspace of Π if it has the property that any line meeting X in at least two points is contained in X . A subspace X together with the lines contained in it is a partial linear space. Subspaces are usualy identified with these partial linear spaces. As the intersection of any collection of subspaces is again a subspace, we can define for each subset X of P the subspace generated by X to be the smallest subspace containing X . This subspace will be denoted by 〈 X 〉. A plane is a subspace generated by two intersecting lines. In [3], [6] partial linear spaces are considered in which all planes are either isomorphic to an affine plane or the dual of an affine plane. Such spaces are called generalized Fischer spaces .