Some remarks on order types and decompositions of sets

This paper is a sequel to the author's previous article on order types [3], to which the reader is referred for all unfamiliar terms and definitions. The two principal problems under investigation here are: (P) to study the existence of order types r such that o <T <E, where E and o, obeing an order type K 7, are given; and (Q) to decompose a set into the union of disjoint sets which have some special properties. ?1 deals with conditions on E and owhich guarantee the existence of at least one order type r in problem (P). In particular, considerable attention is focused on the case where E--aw. In ?2 conditions are stated which ensure the existence of precisely n distinct order types T, where n is any nonnegative integer. Finally, some scattered results pertaining to problem (Q) are given (?3). For example: (1) each linear set of power 2Vo is the union of 2Vo disjoint, exact sets (Theorem 3.1); and (2) if A and B are any two disjoint, similar sets whose union is the set of real numbers, then to each point p of A there corresponds a similarity transformation f, of A into itself for which f,(p) 5 p (Theorem 3.2). 1. Sufficiency conditions for problem (P). As in [3], the set of real numbers is designated by R. X will represent the order type of the real numbers, and `q the rationals ordered in the natural manner. 0 will designate the smallest ordinal number whose power is 2?o.