Affine Barbilian structures

Abstract

W.Leissner has characterized, by geometric axioms, the affine Barbilian planes over a Z-ring (i.e, a ring with 1 such that ab=1 ⇒ ba=1) [10].The aim of the present paper is to characterize correspondingly the affine Barbilian planes over an arbitrary ring with 1. First we shall deal with the translation Barbilian planes, which generalize Leissner's parallelodromic planes [11]. The paper concludes with a study of the kernel of the translation Barbilian plane.

Here, the terms “affine Barbilian structure” and “affine Barbilian plane” are used in a more general sense than in [10] and [11]. Also, the definitions of translation and parallelodromy are slightly different from those in [10] and [11], insomuch that the invariance of the non-neighbour relation is not postulated any more, this being a consequence in a translation Barbilian plane.

In H.J.Arnold's geometry of rings, for any two distinct points, there exists a smallest line incident with them [1]. This property, assumed only for the non-neighbour pairs of points, will replace the usual postulate that two non-neighbour points are incident with exactly one line. Thus, ideas of D.Barbilian [2] and H. J.Arnold [1] are combined with methods of affine ring-geometry due to J.Hjelmslev [5], [6], W.Klingenberg [7], [8], [9], H.Lüneburg [12], W.Benz [3], [4], W.Leissner [10], [11], and others. Many parts of the proofs in [10] and [11] could be used here almost unchanged, under relaxed assumptions.