13 research outputs found
Semiaffine spaces
In this paper we improve on a result of Beutelspacher, De Vito & Lo Re, who characterized in 1995 finite semiaffine spaces by means of transversals and a condition on weak parallelism. Basically, we show that one can delete that condition completely. Moreover, we extend the result to the infinite case, showing that every plane of a planar space with atleast two planes and such that all planes are semiaffine, comes from a (Desarguesian) projective plane by deleting either a line and all of its points, a line and all but one of its points, a point, or nothing
Two character sets in finite linear spaces
A set of type is a set of points of a finite
linear space with the property that each line of the linear space
meets either or points of . In this paper,
sets of type in finite linear spaces with constant point
degree are studied, and some characterization results are given
Semiaffine stable planes
A locally compact stable plane of positive topological dimension will be
called semiaffine if for every line and every point not in there is
at most one line passing through and disjoint from . We show that then
the plane is either an affine or projective plane or a punctured projective
plane (i.e., a projective plane with one point deleted). We also compare this
with the situation in general linear spaces (without topology), where P.
Dembowski showed that the analogue of our main result is true for finite spaces
but fails in general
The fundamental theorem of affine geometry
We deal with a natural generalization of the classical Fundamental Theorem of Affine Geometry to the case of non bijective maps. This extension geometrically characterizes semiaffine morphisms. It was obtained by W. Zick in 1981, although it is almost unknown. Our aim is to present and discuss a simplified proof of this result