Title:

On linearly ordered sets and permutation groups of uncountable degree

In this thesis a set, Ω, of cardinality N_{K} and a group acting on Ω, with N_{K+1} orbits on the power set of Ω, is found for every infinite cardinal N_{K}. Let W_{K} denote the initial ordinal of cardinality N_{K}. Define N := {α_{1}α_{2} . . . α_{n}∣ 0 < n < w, α_{j} ∈ w_{K} for j = 1, . . .,n, α_{n} a successor ordinal} R := {ϰ ∈ N ∣ length(ϰ) = 1 mod 2} and let these sets be ordered lexicographically. The order types of N and R are Κtypes (countable unions of scattered types) which have cardinality N_{K} and do not embed w*_{1}. Each interval in N or R embeds every ordinal of cardinality N_{K} and every countable converse ordinal. N and R then embed every Ktype of cardinality N_{K} with no uncountable descending chains. Hence any such order type can be written as a countable union of wellordered types, each of order type smaller than w^{w}_{k}. In particular, if α is an ordinal between w^{w}_{k} and w_{K+1}, and A is a set of order type α then A= ⋃_{nAn where each An has order type wnk. If X is a subset of N with X and N  X dense in N, then X is orderisomorphic to R, whence any dense subset of R has the same order type as R. If Y is any subset of R then R is (finitely) piece wise orderpreserving isomorphic (PWOP) to R ⋃. Y. Thus there is only one PWOP equivalence class of NKdense Ktypes which have cardinality NK, and which do not embed w*1. There are NK+1 PWOP equivalence classes of ordinals of cardinality N}K. Hence the PWOP automorphisms of R have N_{K+1} orbits on θ(R). The countably piece wise orderpreserving automorphisms of R have N_{0} orbits on R if ∣k∣ is smaller than w_{1} and ∣k∣ if it is not smaller.
