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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?

flag	asked Jun 24 2011 at 16:24 Dima Sustretov 1,287●4●12

John Baldwin · Answer 1 · 2012-07-16 12:40:32Z UTC

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

answered Jul 16 at 12:40

John Baldwin
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Welcome to mathoverflow! – Joel David Hamkins Jul 16 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 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 at 13:58

Sure, but as I have said, such a structure is interpretable in $(M,=)$. Am I missing something? – Dima Sustretov Jul 16 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 at 4:06

a totally categorical structure with trivial geometry which is not interpretable in the trivial structure

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