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A theory $T$ is called near model complete if every formula is equivalent to a Boolean combination of existential formulas mod $T$. I wonder whether there is an equivalent "semantic" definition of this notion like model completeness. (A theory is model complete if every formula is equivalent to an existential formula in that theory or, equivalently, if any embedding of its models is elementary.)

Also, I wonder whether near model completeness of a theory implies its inductiveness or not. Recall that a theory is inductive if the union of any chain (or directed family) of its models is again a model. Equivalently, a theory is inductive iff it is $\forall \exists$-axiomatisable.

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  • $\begingroup$ Something like “in a chain of models of $T$, any formula can change truth value only finitely many times” should work. $\endgroup$
    – Emil Jeřábek
    Oct 18, 2015 at 16:14
  • $\begingroup$ @EmilJeřábek: That was the first step in my failed proof that near model completeness implies inductivity. :) $\endgroup$
    – user12283
    Oct 18, 2015 at 21:43
  • $\begingroup$ @EmilJeřábek: Thanks for your comment. Yes, that may work, but actually I'm looking for a condition that refers only models of the theory and not formulas. $\endgroup$
    – Vahagn Aslanyan
    Oct 18, 2015 at 21:46
  • $\begingroup$ @VahagnAslanyan: Something formula-free might still come out if you pursue this idea further. Maybe something like this: Given $M\models T$, the limit of an ascending chain of substructures of $M$ that are also models of $T$ is again a model of $T$. $\endgroup$
    – user12283
    Oct 18, 2015 at 21:53
  • $\begingroup$ @HansAdler: It's possible that one might get something formula-free out of it, but I think your suggested formulation doesn't work. It's a weaker version of inductiveness which I don't think implies near model completeness. $\endgroup$
    – Vahagn Aslanyan
    Oct 18, 2015 at 22:10

1 Answer 1

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I'm still thinking about your main question. Meanwhile here is my answer to your additional question.

A sandwich criterion by Kueker and Turnquist

You may have seen this before, but it's still worth stating here as it may well be the best answer possible. Proposition 2.3 in Nearly Model Complete Theories by Kueker and Turnquist, MLQ 45 (1999), says that near model completeness is equivalent to the following sandwich condition:

Whenever $M\subseteq N\subseteq M'$ are all models of $T$, $\bar a\in M$ and $(M,\bar a)\equiv (M',\bar a)$, then in fact $(M,\bar a)\equiv(N,\bar a)\equiv(M',\bar a)$.

It's formula-free, but it's not a category theoretic definition.

A nearly model complete theory that is not inductive

Let $T$ be the theory of infinite discrete linear orders, i.e. of non-empty linear orders in which every element has a successor and a predecessor. Define binary existential formulas $<_n$ such that $a<_nb \iff \exists c_1\ldots c_n(a < c_1 < \ldots < c_n < b)$. $T$ clearly has quantifier elimination down to boolean combinations of these formulas. So $T$ is nearly model complete.

For every $n$, $M_n=\frac 1{2^n}\mathbb Z\subset\mathbb Q$ is a model of $T$. However, the union of the chain $(M_n)_{n<\omega}$ consists of the dyadic rationals and is dense, hence not a model of $T$. So $T$ is not inductive.

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  • $\begingroup$ As for the main question, I was thinking that the following could work: whenever $A\subseteq B$ are models of $T$ with $A$ existentially closed in $B$, then $A$ must actually be an elementary substructure of $B$. But this actually defines theories where every formula is equivalent to an $\forall \exists$-formula. Now I think there may not be any nice semantic definition of near model completeness. $\endgroup$
    – Vahagn Aslanyan
    Oct 18, 2015 at 21:35
  • $\begingroup$ Thanks a lot for the reference. I haven't seen that paper before, but observed some of its results myself. However, it contains some other results that are new to me and I think they will be useful for me. I agree that it may be the best possible answer. As I mentioned in my previous comment, I think there is no purely category theoretic definition. Many thanks again for your help. $\endgroup$
    – Vahagn Aslanyan
    Oct 18, 2015 at 23:37
  • $\begingroup$ Do you have an example showing that the following necessary condition isn't actually sufficient? If $M\subseteq N\subseteq M'$ are all models of $T$ and $M\preceq M'$, then $M\preceq N$. $\endgroup$
    – user12283
    Oct 19, 2015 at 8:34
  • $\begingroup$ Yes, I think I do. Your condition characterises 1-model complete theories (in the terminology of the aforementioned paper). Indeed, if $M \subseteq N \subseteq M'$ are models of $T$ with $M \preceq M'$ then $M \subseteq_1 N$. If $T$ is 1-model complete then this implies $M \preceq N$. $\endgroup$
    – Vahagn Aslanyan
    Oct 19, 2015 at 10:06
  • $\begingroup$ Ah, OK. That seems to settle it then. Probably no categorical characterisation. $\endgroup$
    – user12283
    Oct 19, 2015 at 10:26

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