In this thesis a set, Ω, of cardinality N<sub>K</sub> and a group acting on Ω, with N<sub>K+1</sub> orbits on the power set of Ω, is found for every infinite cardinal N<sub>K</sub>. Let W<sub>K</sub> denote the initial ordinal of cardinality N<sub>K</sub>. Define N := {α<sub>1</sub>α<sub>2</sub> . . . α<sub>n</sub>∣ 0 < n < w, α<sub>j</sub> ∈ w<sub>K</sub> for j = 1, . . .,n, α<sub>n</sub> 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 Κ-types (countable unions of scattered types) which have cardinality N<sub>K</sub> and do not embed w*<sub>1</sub>. Each interval in N or R embeds every ordinal of cardinality N<sub>K</sub> and every countable converse ordinal. N and R then embed every K-type of cardinality N<sub>K</sub> 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<sup>w</sup><sub>k</sub>. In particular, if α is an ordinal between w<sup>w</sup><sub>k</sub> and w<sub>K+1</sub>, and A is a set of order type α then A= ⋃<sub>n<w</sub>A<sub>n</sub> where each A<sub>n</sub> has order type w<sup>n</sup><sub>k</sub>. 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 order-preserving isomorphic (PWOP) to R ⋃<sup>.</sup> Y. Thus there is only one PWOP equivalence class of N<sub>K</sub>-dense K-types which have cardinality N<sub>K</sub>, and which do not embed w*<sub>1</sub>. There are N<sub>K+1</sub> PWOP equivalence classes of ordinals of cardinality N</sub>K</sub>. Hence the PWOP automorphisms of R have N<sub>K+1</sub> orbits on θ(R). The countably piece- wise orderpreserving automorphisms of R have N<sub>0</sub> orbits on R if ∣k∣ is smaller than w<sub>1</sub> and ∣k∣ if it is not smaller.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:258020 |
| Date | January 1990 |
| Creators | Ramsay, Denise |
| Contributors | Neumann, P. M. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:ce9a8b26-bb4c-4c85-8231-78e89ce4109d |
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