The poset of copies for automorphism groups of countable relational structures

Abstract

Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak S(U)$ of all permutations of a countable set $U$. Let $\overline{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $\overline{\mathrm{G}}[U]:=\{f[U]\mid f\in \overline{\mathrm{G}}\}$ of images of the functions in $\overline{\mathrm{G}}$, being ordered under inclusion. This set $\overline{\mathrm{G}}[U]$ of subsets of the set $U$ will be called the \emph{poset of copies for} the group $\mathrm{G}$. A denomination being justified by the fact that for every subgroup $\mathrm{G}$ of the symmetric group $\mathfrak S(U)$ there exists a homogeneous relational structure $R$ on $U$ such that $\overline G$ is the set of embeddings of the homogeneous structure $R$ into itself and $\overline{\mathrm{G}}[U]$ is the set of copies of $R$ in $R$ and that the set of bijections $\overline G\cap \mathfrak S(U)$ of $U$ to $U$ forms the group of automorphisms of $\mathrm{R}$.