The poset of copies for automorphism groups of countable relational structures

Abstract

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