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Robert Barham

Publications and source records attributed to Robert Barham.

6 recordsLinked to original sources
arXiv2016

Automatic Homeomorphicity of Locally Moving Clones

We extend the work of M. Rubin on locally moving groups to clones, showing that a locally moving polymorphism clone has automatic homeomorphicity with respect to the class of all polymorphism clones. We show that if $\mathrm{Pol}(M,\mathcal{L})$ is the polymorphism clone of a reduct of $(\mathbb{Q},<)$ or $(\mathbb{L},C)$ such that $\mathrm{Aut}(M,\mathcal{L}) \not= \mathrm{Aut}(M,=)$ and $\mathrm{End}(M,\mathcal{L}) = \mathrm{Emb}(M,\mathcal{L})$ then $\mathrm{Pol}(M,\mathcal{L})$ is locally moving (and hence has automatic homeomorphicity with respect to the class of all polymorphism clones), where $\mathbb{Q}$ is the rationals, $(\mathbb{L},C)$ is the infinite binary-branching homogeneous $C$-relation. We also show that if $\mathcal{M}=(\mathbb{B}, \cup , \cap, \,^c, 1, 0)$, the Fraïssè Generic Boolean algebra, then $\mathrm{Pol}(\mathcal{M})$ is locally moving.

math.LO↗
arXiv2015

The Reconstruction of Cycle-free Partial Orders from their Abstract Automorphism Groups I : Treelike CFPOs

In this triple of papers, we examine when two cycle-free partial orders can share an abstract automorphism group. This question was posed by M. Rubin in his memoir concerning the reconstruction of trees. In this first paper, we give a variety of conditions that guarantee when a CFPO shares an automorphism group with a tree. Some of these conditions are conditions on the abstract automorphism group, while some are one the CFPO itself. Some of the lemmas used have corollaries concerning the model theoretic properties of a CFPO.

math.LO↗
arXiv2015

The Reconstruction of Cycle-free Partial Orders from their Abstract Automorphism Groups II : Cone Transitive CFPOs

In this triple of papers, we examine when two cycle-free partial orders can share an abstract automorphism group. This question was posed by M. Rubin in his memoir concerning the reconstruction of trees. In this middle paper, we adapt a method used by Shelah in \cite{ShelahPermutation} and \cite{ShelahPermutationErrata}, and by Shelah and Truss in \cite{ShelahTrussQuotients} to define a cone transitive CFPO inside its automorphism group using the language of group theory.

math.LO↗
arXiv2015

The Reconstruction of Cycle-free Partial Orders from their Abstract Automorphism Groups III : Decorated CFPOs

In this triple of papers, we examine when two cycle-free partial orders can share an abstract automorphism group. This question was posed by M. Rubin in his memoir concerning the reconstruction of trees. In this final paper, we give describe a way of constructing `decorated' CFPOs by attaching treelike CFPOs to and between the elements of a cone transitive CFPO. We then show that the automorphism groups of the components of of a decorated CFPO are second order definable in the abstract automorphism group of the decorated CFPO.

math.LO↗
arXiv2015

The $\aleph_{0}$-categorical Trees and Cycle-free Partial Orders

We provide a description of the structure of $\aleph_0$-categorical trees and cycle-free partial orders. First the maximal branches of $\aleph_0$-categorical tree are examined, followed by the configuration of the ramification orders, which are then combined to provided necessary and sufficient conditions for a tree to be $\aleph_0$-categorical in terms of these two things. The classification of the $\aleph_0$-categorical cycle-free partial orders is found as a corollary.

math.LO↗