On linearly ordered sets and permutation groups of uncountable degree

Thesis



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 := {α₁α₂ . . . 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 Κ-types (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 K-type 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= ⋃_n<wA_n

where each A_n has order type wⁿ_k.

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 N_K-dense K-types which have cardinality N_K, and which do not embed w*₁. There are N_K+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 N₀ orbits on R if ∣k∣ is smaller than w₁ and ∣k∣ if it is not smaller.

Authors: Ramsay, D; Ramsay, Denise

• Publication date: 1990
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Group theory; Infinite groups