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

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.

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.

传递扩展的通用结构自同构群

Transitive Extensions of Automorphism Groups of Generic Structures

研究传递扩展的存在性，发现仅当颜色数为2的幂时，边缘着色的k超图有传递扩展；k超 tournaments 在Cherlin意义上仅当k为偶数时存在传递扩展。

On the classification of vertex-transitive structures

We consider the classification problem for several classes of countable structures which are "vertex-transitive", meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above $E_0$ in complexity.

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model $\mathcal{M}$ of set theory, and a nontrivial automorphism $j$ of $\mathcal{M}$, let $\mathcal{I}_{\mathrm{fix}}(j)$ be the submodel of $\mathcal{M}$ whose universe consists of elements $m$ of $\mathcal{M}$ such that $j(x)=x$ for every $x$ in the transitive closure of $m$ (where the transitive closure of $m$ is computed within $\mathcal{M}$). Here we study the class $\mathcal{C}$ of structures of the form $\mathcal{I}_{\mathrm{fix}}(j)$, where the ambient model $\mathcal{M}$ satisfies a frugal yet robust fragment of $\mathrm{ZFC}$ known as $\mathrm{MOST}$, and $j(m)=m$ whenever $…

Strongly transitive multiple trees

We give an amalgamation construction of free multiple trees with a strongly transitive automorphism group. The construction shows that any partial codistance function on a tuple of finite trees can be extended to yield multiple trees.