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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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@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 . – Emil Jeřábek Sep 8 '14 at 15:20
    
@EmilJeřábek OK. sorry, I did not attention to these clutter. – Mostafa Mirabi Sep 8 '14 at 20:54
    
Reposted with more details as mathoverflow.net/questions/231791 . – Emil Jeřábek Feb 22 at 9:34

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

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Welcome to mathoverflow! – Joel David Hamkins Jul 16 '12 at 13:12
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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 $f(x,y)=(y,x)$. – Dima Sustretov Jul 16 '12 at 13:23
    
sorry, Neglected to say $f(x) \neq x$ and `there exist infinitely many elements' to get a complete theory. – John Baldwin Jul 16 '12 at 13:58
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Sure, but as I have said, such a structure is interpretable in $(M,=)$. Am I missing something? – Dima Sustretov Jul 16 '12 at 14:48
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@JohnBaldwin Your claim only holds for 1-dimensional interpretations. Dima Sustretov gave above an explicit 2-dimensional interpretation of your theory in $(M,=)$. – Emil Jeřábek Feb 22 at 9:32

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