A Comparison of Algebraic, Metric, and Lattice Betweenness

Introduction. We propose to investigate here the consequences of the identity of each pair chosen from three important generalizations of the relation of betweenness on a line, namely, algebraic betweenness [l, p. 27 J, metric betweenness [3, p. 36], and lattice betweenness [7, Part I I ] . We shall also find an interpretation of metric betweenness in the Banach space of all continuous functions defined on the interval 0^ /^ j 1 which can be used to establish the fact that this relation satisfies no strong four or five point transitivity [7, Part I ] except h and h. We note first that algebraic betweenness implies metric betweenness and lattice betweenness. We find that algebraic betweenness and metric betweenness coincide in a seminormed real vector space if and only if it is strictly convex in the sense of Clarkson [4, p. 404]. We then show that the coincidence of metric and lattice betweenness in a semimetric space [3, p. 38] which is also a lattice [2, p. 16] leads to a system which is a metric lattice (in the sense of G. Birkhoff [2, p. 41]). I t follows that a complete seminormed real vector lattice is equivalent to an (L)-space [ó] if and only if its metric and lattice betweenness relations are identical. Finally, we prove that algebraic and lattice betweenness coincide in a real vector lattice if and only if it is equivalent to the system of all real numbers. We conclude by giving the interpretation of metric betweenness in the space C[0, l ] .