Inter-relations among the Four Principal Types of Order*

The four types of order whose inter-relations are considered in this paper may be called, for brevity, (1) serial order; (2) betweenness; (3) cyclic order; and (4) separation. We first recapitulate the known sets of postulates which define each of these types as an abstract system, and recall the usual geometric interpretation of each type; we then develop the way in which each of these four types may be defined in terms of each of the other three. (For convenience of reference, the numbering of the postulates in earlier publications has been retained.) 1. Serial order. A system (K, R), where if is a class of elements A, B, C, • ■ ■ , and R(AB), or simply AB, is a dyadic relation, is called a "system of serial order" when and only when the following four postulates are satisfied. In each of these postulates it is understood that distinct letters represent distinct elements of K. [The notation " =0" means "is false"; the "horseshoe", d , means "If . . . then"; the "wedge," v, means "or" (in the sense of "at least one"); and the "dot," . , means "and." Dots, singly or in groups, serve also as punctuation marks.]