This paper deals with neighbor-preserving epimorphisms between arbitrary projective Klingenberg planes. Our main result is an algebraic characterization of such epimorphisms which generalizes a theorem of J. C. Ferrar and F, D. Veldkamp [2] for neighbor-preserving epimorphisms between projective ring planes 0. INTRODUCTION Homomorphisms between projective planes were first studied by W. Klingenberg I-6]. He has given an algebraic description of epimorphisms between Desarguesian projective planes by means of mappings whose domain is a valuation ring of the coordinatizing (skew) field. L. Skornyakov I-7] and J. Andr6 I-1] have given an algebraic description of epimorphisms between ordinary projective planes by means of certain mappings, called places, between the coordinatizing PTRs.
It is natural to ask for a generalization of the results to projective Klingenberg planes. These planes consist of points and lines together with an incidence and a neighbor relation. An epimorphism should preserve the incidence relation but it is not clear what it should do with the neighbor relation. In this paper we study epimorphisms between arbitrary PK-planes which preserve the neighbor relations between points and lines respectively. The Desarguesian case has already been considered by G. T6rner in 1,8] and by J. C. Ferrar and F. D. Veldkamp in 1,2]. Our main result is an algebraic characterization of neighbor-preserving epimorphisms between PK-planes. Therefore we use the coordinatization of such planes by planar sexternary rings, as treated in I-4].
In a forthcoming paper 15] we study epimorphisms between PK°planes which preserve the non-neighbor relation (distant-preserving epimorphisms.). 1. NEIGHBOR-PRESERVING EPIMORPHISMS BETWEEN PK-PLANES 1.1. A projective Klingenberg plane (PK-plane) is an incidence structure with neighbor relation X = (P, B, I, ,-~), ~ = (~e, ~B) (P = point-set, B = lineset, I = symmetric incidence relation, H e (resp. ~ B) = equivalence relation in P (resp. B) called neighbor relation) which satisfies the following axioms: