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Title: On linearly ordered sets and permutation groups of uncountable degree
Author: Ramsay, Denise
ISNI:       0000 0001 3507 0074
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 1990
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In this thesis a set, Ω, of cardinality NK and a group acting on Ω, with NK+1 orbits on the power set of Ω, is found for every infinite cardinal NK. Let WK denote the initial ordinal of cardinality NK. Define N := {α1α2 . . . αn∣ 0 < n < w, αj ∈ wK for j = 1, . . .,n, αn 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 NK and do not embed w*1. Each interval in N or R embeds every ordinal of cardinality NK and every countable converse ordinal. N and R then embed every K-type of cardinality NK 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 wwk. In particular, if α is an ordinal between wwk and wK+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 order-preserving isomorphic (PWOP) to R ⋃. Y. Thus there is only one PWOP equivalence class of NK-dense K-types which have cardinality NK, and which do not embed w*1. There are NK+1 PWOP equivalence classes of ordinals of cardinality NK. Hence the PWOP automorphisms of R have NK+1 orbits on θ(R). The countably piece- wise orderpreserving automorphisms of R have N0 orbits on R if ∣k∣ is smaller than w1 and ∣k∣ if it is not smaller.
Supervisor: Neumann, P. M. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Infinite groups ; Group theory