Generic expansions of ω-categorical structures and semantics of generalized quantifiers

Abstract

Let M be a countably infinite ω-categorical structure. Consider Aut(M) as a complete metric space by defining d(g, h) = Ω{2−n: g (xn) ≠ h(xn) or g−1 (xn) ≠ h−1 (xn)} where {xn : n ∈ ω} is an enumeration of M An automorphism α ∈ Aut(M) is generic if its conjugacy class is comeagre. J. Truss has shown in [11] that if the set P of all finite partial isomorphisms contains a co-final subset P1 closed under conjugacy and having the amalgamation property and the joint embedding property then there is a generic automorphism. In the present paper we give a weaker condition of this kind which is equivalent to the existence of generic automorphisms. Really we give more: a characterization of the existence of generic expansions (defined in an appropriate way) of an ω-categorical structure. We also show that Truss' condition guarantees the existence of a countable structure consisting of automorphisms of M which can be considered as an atomic model of some theory naturally associated to M. We do it in a general context of weak models for second-order quantifiers.The author thanks Ludomir Newelski for pointing out a mistake in the first version of Theorem 1.2 and for interesting discussions. Also, the author is grateful to the referee for very helpful remarks.