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

Heap - ternary algebraic structure

In this paper some classes of ternary algebraic structures (semi-heaps, heaps) are considered. The connection between heaps (laterally commutative heaps) and corresponding algebraic and geometric structures is presented. The equivalence of heap existence and the Desargues system on the same set is directly proved. It is the starting point for an analogous result about a laterally commutative heap and a parallelogram space

Semiaffine stable planes

A locally compact stable plane of positive topological dimension will be called semiaffine if for every line $L$ and every point $p$ not in $L$ there is at most one line passing through $p$ and disjoint from $L$. 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