Thank you, John, for your comment and mentions (of Borovik-Nesin)…

Andrés

]]>“The notion of interpretation in model theory corresponds to a number of familiar phenomena in algebra which are often considered distinct: coordinatization, structure theory, and constructions like direct product and homomorphic image. For example a Desarguesian projective plane is coordinatized by a division ring; Artinian semisimple rings are finite direct products of matrix rings over divisions rings; many theorems of finite group theory have as their conclusion that a certain abstract group belongs to a standard family of matrix groups over . All of these examples have a common feature: certain structures of one kind are somehow encoded in terms of structures of another kind. All of these examples have a further feature which plays no role in algebra but which is crucial for us: in each case the encoded structures can be recovered from the encoding structures definably.”

]]>Now it’s clear, thanks.

]]>@Artem: Shelah does, in fact, have methods to produce many models of arbitrary degrees of saturation in unstable theories, using a somewhat strange notion of isolation. The relevant reference is Theorem VIII.3.2 of his Classification Theory.

]]>@Artem: But the way I define the new theory T’, the predicate P_1 is *not* an elementary submodel! Really, one should think of models of T’ as having two disjoint sorts, P_1 and P_2 (for some reason I thought it would be better in my post to avoid talking about multi-sorted logic…). The sort of P_1 is a model of T, the sort of P_2 has no definable structure other than what is trivially induced by the “many-to-one” projection map pi from P_2 onto the group G. (I’ll edit the definition of T’ in the post now to explain this, since I see that I didn’t explain clearly enough what I had in mind.)

Now unless I am missing something silly, there should be no way to define an infinite linear ordering in either of the sorts P_1 or P_2, unless there was already a way to define an ordering in the original theory T… Basically, if the theory T has q.e., then T’ will have q.e. as well, and then it’s not hard to check what I’m claiming.

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