Catharine A. Baker

Graphs with the n-e.c. adjacency property constructed from affine planes

We give new examples of graphs with the n-e.c. adjacency property. Few explicit families of n-e.c. graphs are known, despite the fact that almost all finite graphs are n-e.c. Our examples are collinearity graphs of certain partial planes derived from affine planes of even order. We use probabilistic and geometric techniques to construct new examples of n-e.c. graphs from partial planes for all n, and we use geometric techniques to give infinitely many new explicit examples if n=3. We give a new construction, using switching, of an exponential number of non-isomorphic n-e.c. graphs for certain orders.

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.

ORDER AND TOPOLOGY IN PROJECTIVE HJELMSLEV PLANES

The concept of an ordered projective Hjelmslev plane was intuitively introduced by Hjelmslev in “Einleitung in die allgemeine Kongruenglehre” ([9], [10]).This paper is concerned with formalizing and examing preorderings and orderings for projective Hjelmslev planes. In addition we show that orderings generated topologies of the point and line sets which render the plane a topological Hjelmslev plane ([19], [13]). These planes — unlike the ordinary ordered planes ([18]) — are, due to the existence of infinitesimals, non-archimedian, non-compact and disconnected with the neighbour classes as certain quasi-components.

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.

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.

Preordered uniform Hjelmslev planes

In “Order and topology in projective Hjelmslev planes”, the authors introduced the notions of preorder and order on projective Hjelmslev planes (PH-planes); cf. [3: 3.7, 5.3]. They also showed that in both ordered and preordered uniform PH-planes singular segments are uniquely determined by their endpoints. In this note, we show that the notions of order and preorder actually coincide on uniform PH-planes.