Modeles Satures et Modeles Engendres par des Indiscernables
In the early eighties, answering a question of A. Macintyre, J. H. Schmerl () proved that every countable recursively saturated structure, equipped with a function $\beta$ encoding the finite functions, is the $\beta$-closure of an infinite indiscernible sequence. This result implies that every countably saturated structure, in a countable but not necessarily recursive language, is an Ehrenfeucht-Mostowski model, by which we mean that the structure expands, in a countable language, to the Skolem hull of an infinite indiscernible sequence (in the new language). More recently, D. Lascar () showed that the saturated model of cardinality $\aleph_1$ of an $\omega$-stable theory is also an Ehrenfeucht-Mostowski model. These results naturally raise the following problem: which (countable) complete theories have an uncountably saturated Ehrenfeucht-Mostowski model. We study a generalization of this question. Namely, we call ACI-model a structure which can be expanded, in a countable language L', to the algebraic closure (in L') of an infinite indiscernible sequence (in L'). And we try to characterize the $\lambda$-saturated structures which are ACI-models. The main results are the following. First it is enough to restrict ourselves to $\aleph_1$-saturated structures: if T has an $\aleph_1$-saturated ACI-model then, for every infinite $\lambda$, T has a $\lambda$-saturated ACI-model. We obtain a complete answer in the case of stable theories: if T is stable then the three following properties are equivalent: (a) T is $\omega$-stable, (b) T has a $\aleph_1$saturated ACI-model, (c) every saturated model of T is an Ehrenfeucht-Mostowski model. The unstable case is more complicated, however we show that if T has an $\aleph_1$-saturated ACI-model then T doesn't have the independence property.