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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 at 15:20
    
@EmilJeřábek OK. sorry, I did not attention to these clutter. –  Mostafa Mirabi Sep 8 at 20:54

1 Answer 1

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
    
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
    
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
    
Yes, interpreted in (M,=) means there must be formula in only = that defines the function f. An easier way to see the example might to write it as equivalence relation so that each class has size 2. No formula of equality can define that equivalence relation. –  John Baldwin Jul 20 '12 at 4:06

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