Endomorphisms of Stable Planes

Abstract

Endomorphisms of stable planes are introduced, and it is shown that these are injective, locally constant or collapsed. Examples are studied, and it is shown that there are stable planes admitting \substantially more" endomorphisms than automorphisms. It seems to be a rather popular feeling that geometry is the domain of groups and symmetry, whereas asymmetry and semigroups have been banished to other elds in mathematics. The purpose of this note is to show that this feeling is justiied in the area of locally compact connected aane or projective planes, but that the situation is completely diierent if one studies hyperbolic planes and their generalizations. Deenition 1. a) A linear space (P; L) consists of a non-empty set P (\points") and a system L of subsets of P (\lines") such that-for each pair (p; q) of distinct points there is exactly one set L 2 L (denoted by pq) such that fp;qg L,-each member of L contains at least two elements, and P 6 2 L. b) A stable plane 1 (P; L) is a linear space whose point space P and line space L are equipped with non-discrete 2 Hausdorr 3 topologies such that-the join map _:P 2 n f(p;p)jp 2 Pg ! L:(p;q) 7 ! pq is continuous,-the set D :=

Cite this paper

@inproceedings{Stroppel1992EndomorphismsOS, title={Endomorphisms of Stable Planes}, author={Markus J. Stroppel}, year={1992} }