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Among the theorems of early geometric model theory there is one by Lachlan stating that every totally categorical structure with a trivial pregeometry is intrepretable in a dense linear order.

That suggests in particular that there are totally categorical structures with a trivial pregeometry that are not interpretable in the trivial structure $(M,=)$. Could someone kindly give an example of such a structure?

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    $\begingroup$ @MostafaMirabi Please do not make trivial edits to old posts, especially in such mass numbers. These clutter up the main page. This has been discussed a few times on the meta, see e.g. meta.mathoverflow.net/questions/599 . $\endgroup$ Sep 8, 2014 at 15:20
  • $\begingroup$ @EmilJeřábek OK. sorry, I did not attention to these clutter. $\endgroup$ Sep 8, 2014 at 20:54
  • $\begingroup$ Reposted with more details as mathoverflow.net/questions/231791 . $\endgroup$ Feb 22, 2016 at 9:34

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One example is the theory of a unary functions satisfying $f^2(x) =x$.

As pointed out below by Eric, (with the explicit example of someone else) I was thinking only of 1-dimensional interpretations.

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    $\begingroup$ Welcome to mathoverflow! $\endgroup$ Jul 16, 2012 at 13:12
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    $\begingroup$ Could you tell a bit more precisely what structure you have in mind? For example, if for all $x$, $f(x)\neq x$, then this is interpetable in $(M,=)$ with the universe defined by $x \neq y$ and $f(x,y)=(y,x)$. $\endgroup$ Jul 16, 2012 at 13:23
  • $\begingroup$ sorry, Neglected to say $f(x) \neq x$ and `there exist infinitely many elements' to get a complete theory. $\endgroup$ Jul 16, 2012 at 13:58
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    $\begingroup$ Sure, but as I have said, such a structure is interpretable in $(M,=)$. Am I missing something? $\endgroup$ Jul 16, 2012 at 14:48
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    $\begingroup$ @JohnBaldwin Your claim only holds for 1-dimensional interpretations. Dima Sustretov gave above an explicit 2-dimensional interpretation of your theory in $(M,=)$. $\endgroup$ Feb 22, 2016 at 9:32
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An example of such a structure is given here: ω-categorical, ω-stable structure with trivial geometry not definable in the pure set. Essentially, it is a non-split cover of the theory of the pure set. As in the paper of Hrushovski on Totally Categorical Structures.

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