Further results on order types and decompositions of sets

Abstract

In this paper the study of problems P and Q [3; 4] is continued. The reader is referred to the aforementioned two references for all unfamiliar terms and symbols. The first three sections deal with the decomposition of a linear set into a family of pairwise disjoint sets, the order types of the sets being required to satisfy some specified conditions. ?1 is concerned with the decomposition of an arbitrary linear set, of power 2H0, into a family of pairwise disjoint sets, the order types of the sets being pairwise incomparable. ?2 is concerned with the decomposition of certain linear sets into families of pairwise disjoint, similar sets. ?3 is concerned with the decomposition of an arbitrary linear set into families, of power No and 2V0, of pairwise disjoint sets, each set having property A. Let {I }, t <0, be a sequence of order types, of power 2 o each, such that at <X for each t. In ?4 it is shown that (1) problem P, as applied to ot and , =X, admits of a solution rt such that the rt are pairwise incomparable order types (Theorem 4.1); and (2) problem P, as applied to each o= 0 and /j = at, admits of a solution rt such that the rt are pairwise incomparable order types (Theorem 4.3). 1. Decompositions into incomparable order types. In this section the decomposition of a linear set into a finite number and into a denumerably infinite number of pairwise disjoint sets Ai, where the order types of the Ai are pairwise incomparable, is studied. The decomposition of a linear set into 2Ho pairwise disjoint sets Ai, where the order types of the Ai are pairwise incomparable, is treated in ?3 (Theorem 3.4). DEFINITION. A linear set E, of power 2H0, will be said to have property C if each element of E is a c-condensation point of E. For any linear set E, by K(E) is meant the set of similarity transformations of E into R. By K*(E) is meant the set K(E) {I}, where I is the identity transformation of E. For any similarity transformation f of A into B, by f* is meant the inverse of f.