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) over bar be the topological closure of G in the function topology on U-U. We initiate the study of the poset (G) over bar [U] := {f [U] | f is an element of (G) over bar} of images of the functions in (G) over bar, being ordered under inclusion. This set (G) over bar [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) over bar is the set of embeddings of the homogeneous structure R into itself and (G) over bar [U] is the set of copies of R in R and that the set of bijections (G) over bar boolean AND S(U) of U to U forms the group of automorphisms of R.