Preordered affine Hjelmslev planes

An affine Hjelmslev plane (AH-plane)H is preordered if it is endowed with a betweenness relation which is preserved by bijective parallel projections. Preordered biternary rings are defined and are used to construct preordered AH-planes. Conversely, a preordered AH-plane induces a preordering on any of its biternary rings. The concepts used in this paper are compared with those in a recent independent study of preordered affine Klingenberg planes by F. Machala. Unlike the Klingenberg case, preordered AH-planes always possess convex neighbour classes and unlike ordered ordinary affine planes, preordered AH-planes are never archimedian.The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.     

Coordinate rings of topological Klingenberg planes II: the algebraic foundation for a projective theory

Ternary fields are the coordinate rings of affine and projective planes; however, the planes constructed over topological ternary fields are not necessarily topological. Surprisingly, the explanation of this phenomenon becomes evident in the more general theory of topological Klingenberg planes as we exhibited in [3] for the affine case. However, in the projective setting, we have a more formidable task. We must develop a new coordinate ring that admits a topological structure suitable for coordinatizing topological PK-planes. We accomplish this in two stages. In this paper, we revisit the standard coordinate rings [1, 11], discuss and resolve their deficiencies by developing a new coordinate ring as a unique extension of these refined standard rings. In a subsequent paper [4], we show that this new ring can be suitably topologized to coordinatize a topological PK-plane. This last result can then be used to explain why topological ternary fields do not necessarily coordinatize topological projective planes. Received 17 February 2000; revised 10 June 2000.     

Ordered affine Hjelmslev planes

Preorderings and orderings of affine Hjelmslev planes were defined in [3]. The ordering of an AH-plane is shown to induce an ordering on each coordinate biternary ring which is compatible with the multiplication of the associated algebra. Even stronger projective order relations, on both the planes and biternary rings, are introduced and are shown to be also compatible with the ternary operations.The author gratefully acknowledges the support of the Natural Sciences and Engineering Research Council of Canada, and the invaluable assistance of Dr. N. D. Lane, McMaster University.     

Elementary constructions of locally compact affine planes

In this paper we describe several elementary constructions of 4-, 8- and 16-dimensional locally compact affine planes. The new planes share many properties with the classical ones and are very easy to handle. Among the new planes we find translation planes, planes that are constructed by gluing together two halves of different translation planes, 4-dimensional shift planes, etc. We discuss various applications of our constructions, e.g. the construction of 8- and 16-dimensional affine planes with a point-transitive collineation group which are neither translation planes nor dual translation planes, the proof that a 2-dimensional affine plane that can be coordinatized by a linear ternary field with continuous ternary operation can be embedded in 4-, 8- and 16-dimensional planes, the construction of 4-dimensional non-classical planes that admit at the same time orthogonal and non-orthogonal polarities. We also consider which of our planes have tangent translation planes in all their points. In a final section we generalize the Knarr-Weigand criterion for topological ternary fields.This research was supported by a Feodor Lynen fellowship.     

Laguerre and Minkowski planes with neighbor relation

In [7] we have introduced the notion of a Möbius plane with neighbor relation as a generalization of ordinary Möbius planes. In this paper we present two other classes of circle geometries which are locally affine Klingenberg planes: Laguerre and Minkowski planes with neighbor relation.Research supported by IWONL grant no-840037     

Smooth Stable Planes and the Moduli Spaces of Locally Compact Translation Planes

 Smooth stable planes have been introduced in [3]. At every point p of a smooth stable plane the tangent spaces of the lines through p form a compact spread (see the definition in Section 2) on the tangent space thus defining a locally compact topological affine translation plane . We introduce the moduli space of isomorphism classes of compact spreads, . We show that for the topology of is not by constructing a sequence of non-classical spreads in that converges to the classical spread in , where . Moreover, we prove that the isomorphism type of varies continuously with the point p. Finally, we give examples of smooth affine planes which have both classical and non-classical tangent translation planes. (Received 15 April 1999; in revised form 22 October 1999)     

Affine Klingenbergsche Strukturen

In this paper we define the affine Klingenberg structures (AK-structures) as a generalization of the affine Klingenberg planes (AK-planes) [2]. By means of moduls over the local rings (which need not be free as with the coordinate AK-planes is the case) there is constructed a class of the coordinate AK-structures. In II and III we define the Desarguesian AK-structures. Any coordinate AK-structure is a Desarguesian AK-structure. With the methods established by E.Artin and applied by W.Klingenberg in [4] we show that any Desarguesian AK-structure is isomorphic with the coordinate AK-structure. Some of the results obtained have been applied to the theory of the AK-planes. Thus, for example, it has been shown that it is possible to assume less with the definition of the Desarguesian AK-planes than with [4] or even with [5].     

Die Oktavenebene als Translationsebene mit großer Kollineationsgruppe

It is shown that the affine plane over the Cayley numbers is the only 16-dimensional locally compact topological translation plane having a collineation group of dimension at least 41. This (hitherto unpublished) result is one of the ingredients of H. Salzmann's characterizations of the Cayley plane among general compact projective planes by the size of its collineation group.The proof involves various case studies of the possibilities for the structure and size of collineation groups of 16-dimensional locally compact translation planes. At the same time, these case studies are important steps for a classification program aiming at the explicit determination of all such translation planes having a collineation group of dimension at least 38.     

Cartesian groups not belonging to topological projective planes

A new class of topological Cartesian groups is introduced by bending the lines of the affine coordinate plane over an ordered skewfield countably infinitely often. These Cartesian groups belong to topological affine planes with continuous parallelism, but they need not yield a topological projective plane if the topology is not equal to the order topology of the skewfield, in contrast to the results of HARTMANN [6] for generalized Moulton planes. Dedicated to Professor Dr. Gerhard Grimeisen on his 70 ^th birthday     

On 4-dimensional Minkowski planes with 7-dimensional automorphism groups

This paper concerns 4-dimensional (topological locally compact connected) Minkowski planes that admit a 7-dimensional automorphism group . It is shown that such a plane is either classical or has a distinguished point that is fixed by the connected component of and that the derived affine plane at this point is a 4-dimensional translation plane with a 7-dimensional collineation group.This research was supported by a Feodor Lynen Fellowship.     

Unitals in Topological Affine Translation Planes Need Not Be Strictly Convex

We study the relationship of two incidence geometric convexity notions, namely, ovoids in real affine spaces and compact unitals of codimension 1 in topological affine translation planes. In [3] we showed that every ovoid in a translation plane is a unital, and we asked if the converse is true. Here we introduce the notion of a shell, which is distinctly weaker than that of an ovoid and still implies the unital property if the translation plane is properly chosen (and the shell is not too degenerate). We give an explicit example of a shell that is not an ovoid. The question remains whether or not conversely, every compact unital of codimension 1 in a translation plane is a shell.     

Semiclassical Topological MöBius Planes

A new rather large family of locally compact 2-dimensional topological Mobius planes is introduced here. This family consists exactly of those Mobius planes which can be obtained by pasting together two halves of the classical real Mobius plane suitably. Isomorphism classes, automorphisms, and the Hering type of these planes are determined.     

Semiclassical topological flat Laguerre planes obtained by pasting along two parallel classes

A new rather large family of locally compact 2-dimensional topological Laguerre planes is introduced here. This family consists exactly of those Laguerre planes which can be obtained by pasting together two halves of the classical real Laguerre plane along two parallel classes suitably. Isomorphism classes and automorphisms of these planes are determined.     

Semiclassical Topological Flat Laguerre Planes Obtained by Pasting Along A Circle

A new rather large family of locally compact 2-dimensional topological Laguerre planes is introduced here. This family consists exactly of those Laguerre planes which can be obtained by pasting together two halves of the classical real Laguerre plane along a circle suitably. Isomorphism classes and automorphism groups of these planes are determined. Together with [9] this gives a complete classification of all semicalssical topological flat Lguerre planes.     

Unitals in Topological Affine Translation Planes Need Not Be Strictly Convex

We study the relationship of two incidence geometric convexity notions, namely, ovoids in real affine spaces and compact unitals of codimension 1 in topological affine translation planes. In [3] we showed that every ovoid in a translation plane is a unital, and we asked if the converse is true. Here we introduce the notion of a shell, which is distinctly weaker than that of an ovoid and still implies the unital property if the translation plane is properly chosen (and the shell is not too degenerate). We give an explicit example of a shell that is not an ovoid. The question remains whether or not conversely, every compact unital of codimension 1 in a translation plane is a shell.This paper was written while the third author was supported by a grant from DFG and TÜBITAK.Received March 12, 2002 Published online November 18, 2002     

An Affine Characterization of Moufang Projective Klingenberg Planes

It is shown that with minor exceptions all projective Klingenberg planes are Moufang exactly if they have a trilateral each of whose sides is the line at infinity of a translation plane; but there are non-Moufang projective Klingenberg planes which have a single line which produces an equiaffine translation plane or a point P so that all lines through P, except perhaps for some neighbour lines, produce translation planes.     

4-Dimensional Elation Laguerre Planes Admitting Non-Solvable Automorphism Groups

 This paper concerns 4-dimensional (topological locally compact connected) elation Laguerre planes that admit non-solvable automorphism groups. It is shown that such a plane is either semi-classical or a single plane admitting the group SL(2, ). Various characterizations of this single Laguerre plane are obtained. Received October 17 2000; in revised form April 23 2001 Published online August 5, 2002     

Minimal topological projective planes

The atoms of the lattice of compatible topologies on a given projective plane are examined. The notion of a field of type V is generalized to ternary fields of type V. These are always minimal, and they arise for example as coordinate structure of strictly uniformizable or of orderable topological planes. Hence, such planes are minimal.Dedicated to Professor Helmut Karzel on his 60^th birthday     

A note on Hilbert and Beltrami systems

Hilbert and Beltrami (line- ) systems were introduced by H. Mohrmann, Math. Ann. 85 (1922) p.177- 183. These systems give examples of non- desarguesian affine planes, in fact, the earliest known examples are of this type. We describe a construction for “generalized Beltrami systems”, and show that every such system defines a topological affine plane with point set ?². Since our construction uses only the topological structure of ?²- planes, it is possible to iterate this process. As an application, we obtain an embeddability theorem for a class of two- dimensional stable planes, including Strambach’s exceptional SL₂R- plane.     

Affine planes over finite rings,a summary

In Keppens (Innov. Incidence Geom. 15: 119–139, 2017) we gave a state of the art concerning “projective planes” over finite rings. The current paper gives a complementary overview for “affine planes” over rings (including the important subclass of desarguesian affine Klingenberg and Hjelmslev planes). No essentially new material is presented here but we give a summary of known results with special attention to the finite case, filling a gap in the literature.