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The Poset of Copies for Automorphism Groups of Countable Relational Structures
Claude Laflamme, Maurice Pouzet, Norbert Sauer and Robert Woodrow

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 [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.

Keywords: Homogeneous relational structures, automorphism groups, homogeneity, posets

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