The poset of copies for automorphism groups of countable relational structures

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 UU. 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. © 2021 Old City Publishing. All rights reserved.