Reconstructing Structures with the Strong Small Index Property up to Bi-Definability

2017/03/30 by Paolini, Gianluca, Shelah, Saharon · 1 citation
#03C15 #03C35 #20B27 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1703.10498

Abstract

Let K be the class of countable structures M with the strong small index property and locally finite algebraicity, and K_* the class of M ∈ K such that acl_M(\ a \) = \ a \ for every a ∈ M. For homogeneous M ∈ K, we introduce what we call the expanded group of automorphisms of M, and show that it is second-order definable in Aut(M). We use this to prove that for M, N ∈ K_*, Aut(M) and Aut(N) are isomorphic as abstract groups if and only if (Aut(M), M) and (Aut(N), N) are isomorphic as permutation groups. In particular, we deduce that for ℵ₀-categorical structures the combination of strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin's well-known ∀ ∃-interpretation technique of [7]. Finally, we show that every finite group can be realized as the outer automorphism group of Aut(M) for some countable ℵ₀-categorical homogeneous structure M with the strong small index property and no algebraicity.

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