THE POSET OF COPIES FOR AUTOMORPHISM GROUPS OF COUNTABLE RELATIONAL STRUCTURES

Claude Laflamme , Pouzet Maurice (1, 2) , Norbert Sauer (2) , Robert Woodrow (2)

1 ICJ - Institut Camille Jordan
2 University of Calgary

Let G be a subgroup of the symmetric group S(U)of all permutations of a countable set U. Let \overline G be the topological closure of G in the function topology on set set of maps from U into U. We initiate the study of the poset G[U] ∶= {f[U] ∣ f ∈ G} of images of the functions in \overline G, being ordered under inclusion. This set \overline 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 \overline G is the set of embeddings of the homogeneous structure R into itself and \overline G[U] is the set of copies of R in R and that the set of bijections \overline G∩S(U) of U to U forms the group of automorphisms of R.

Domaines: Mathématique discrète [cs.DM]; Mathématiques [math]; Combinatoire [math.CO]; Logique [math.LO]

Claude Laflamme, Pouzet Maurice, Norbert Sauer, Robert Woodrow. THE POSET OF COPIES FOR AUTOMORPHISM GROUPS OF COUNTABLE RELATIONAL STRUCTURES. Journal of Multiple-Valued Logic and Soft Computing, In press, dedicated to the memory of Ivo Rosenberg, vol 1,, ⟨10.48550/arXiv.2002.04771⟩. ⟨hal-02512435⟩