The reconstruction of cycle-free partial orders from their automorphism groups

Is a cycle-free partial order recognisable from its abstract automorphism group? This thesis resolves that question for two disjoint families: those cycle-free partial orders which share an automorphism group with a tree; and those which satisfy certain transitivity conditions, before giving a method for combining the two. Chapter 1, the introduction, as well as introducing some notation and defining the cyclefree partial orders (CFPOs), gives a list of the results that this thesis calls upon. Chapter 2 gives a structure theorem for ℵ0-categorical trees, which is of particular interest here as their reconstruction problem is completely solved, and for the ℵ0- categorical CFPOs, which when combined with the results in Chapter 3, gives a complete reconstruction result for ℵ0-categorical CFPOs. Chapter 3 asks which CFPOs have an automorphism group isomorphic to one of a tree. It gives conditions on the CFPO and the automorphism group that allow the invocation of the work done by Rubin on the reconstruction of trees. In a brief epilogue the results are also used to show that many of the model theoretic properties of the trees are also properties of the CFPOs. The second family is addressed in Chapter 4 using a method used by Shelah and Truss on the symmetric groups of cardinals, which uses the alternating group on five elements. In Chapter 5 I give a method of attaching structures of the first kind to structures of the second, which admits a second order definition in the abstract automorphism group of the automorphism groups of the components. The last chapter is a discussion of how the work done here can be made more complete. I have included an appendix, which lists the formulas used in Chapters 4 and 5, which the reader can tear out and keep at hand to save flicking between pages

605 revision:1998-05-19 modified:1998-05-19 On Distinguishing Quotients of Symmetric Groups

A study is carried out of the elementary theory of quotients of symmetric groups in a similar spirit to [10]. Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S(µ) on an infinite cardinal µ are all of the form Sκ(µ) = the subgroup consisting of elements whose support has cardinality < κ, for some κ ≤ µ +. A many-sorted structure Mκλµ is defined which, it is shown, encapsulates the first order properties of the group Sλ(µ)/Sκ(µ). Specifically, these two structures are (uniformly) bi-interpretable, where the interpretation of Mκλµ in Sλ(µ)/Sκ(µ) is in the usual sense, but in the other direction is in a weaker sense, which is nevertheless sufficient to transfer elementary equivalence. By considering separately the cases cf(κ)> 2 ℵ0, cf(κ) ≤ 2 ℵ0 < κ, ℵ0 < κ < 2 ℵ0, and κ = ℵ0, we make a further analysis of the first order theory of Sλ(µ)/Sκ(µ), introducing many-sorted second order structures N 2 κλµ, all of whose sorts have cardinality at most 2 ℵ0, and in terms of which we can completely characterize the elementary theory of the groups Sλ(µ)/Sκ(µ).