Transitivities of Betweenness

Introduction. The examination of the foundations of geometry which interested many prominent mathematicians about the turn of the century brought to light the importance of the fundamental notion of betweenness (see, for example('), [10, 11]). This notion has suffered the treatment which modern mathematics metes out to all its concepts, namely, first an examination of the concept in a particular instance followed by wider and wider generalizations. The first part of this program for the concept of betweenness was carried through by Pasch, Huntington and Kline [8, 10]. The simplicity of the concept permitted them to give an elegant and complete theory for the case of linear order. In the direction of generalizations(2), K. Menger and his students have been one of the most important influences in the study of betweenness in metric spaces [9, 3 ]. We purpose here to add to both phases of this program. The first part of our paper continues the analysis of Huntington and Kline into an examination of postulates involving five points; the second part deals mainly with a definition of betweenness in lattices which generalizes metric betweenness in metric lattices (see [5, 6]). It is hoped that the five point transitivities may prove interesting and their analysis valuable. If we restrict our attention to the relation of betweenness in linear order such cannot be the case since four point properties are then sufficient to describe completely the betweenness relation. We feel that the results of the second part exhibit the properties of the betweenness relation as reflections of properties of the underlying space(3). We shall use the notations of set theory which have become standard. In the second part we shall assume a knowledge of the fundamentals of both lattice theory and metric geometry. We refer the reader to the recent books Distance Geometry by L. M. Blumenthal [3] and Lattice Theory by Garrett Birkhoff [1]. We shall use the terminology and notation of these books in the second part.