Ordered Hjelmslev Planes

Abstract

<p>There are two equivalent ways to define order on ordinary affine planes; however generalizations of these definitions to A.H. planes yield two distinct definitions. We investigate the relationship between ordered A.H. planes and their ordered coordinate biternary rings. We introduce two new order relations: projective orderings of A.H. planes which are shown to be equivalent to strong orderings of the coordinate biternary rings of these planes and almost-strong orderings of biternary rings which are equivalent to strong orderings of the corresponding A.H. planes. In addition, we extend the axioms of order for projective planes to P.H. planes and discuss the properties of these order relations.</p> <p>We now show that an A.H. plane embedded in an ordered P.H. plane is itself ordered.</p> <p>We consider the projective completions constructed by Artmann, coordinatize them by means of biternary rings with additional ternary operations and prove various properties of the new ternary operators. We then show that although there exist strongly ordered projectively uniform A.H. planes which do not have ordered projective completions, we can always construct ordered projective completions of projectively ordered projectively uniform A.H. planes.</p>