Briefly, my question is the following.


*does every countable ω-categorical, ω-stable structure with 
disintegrated strongly minimal sets interpret in the countable pure set?*


By countable pure set I mean a structure with countable universe and equality relation only.


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This is a repetition of this question [A totally categorical structure with trivial geometry which is not interpretable in the trivial structure](https://mathoverflow.net/questions/68748/a-totally-categorical-structure-with-trivial-geometry-which-is-not-interpretable). However, I do not understand why the answer provided there is marked correct. (I agree with Dima Sustretov in the comments that the structure does interpret in the pure set).


The background to my question is the following.


It is shown in the paper of Cherlin, Harrington and Lachlan that
every ω-categorical, ω-stable structure is coordinatized by a collection of projective spaces, affine spaces and pure sets (which appear as strictly minimal sets in the expansion of the original structure by imaginaries). I'm interested in those structures in which only the pure sets appear. These were studied in the paper of Lachlan titled "Structures coordinatized by indiscernible sets". In this paper, it is shown that every such structure interpret in an arbitrary countable linear order,
and also, that such structures correspond precisely to reducts of totally categorical structures with trivial geometry (of the strongly minimal sets).


It is easy to see that every structure which interprets in the pure set is 
ω-categorical and ω-stable, and it follows from the paper of Lachlan that it is coordinatized by indiscernible sets (this even shown for structures which interpret in a dense linear order). 


Therefore, we have the following implications:


*interprets in countable pure set → ω-categorical, ω-stable, with disintegrated strongly minimal sets → interprets in countable dense linear order.*


The second implication cannot be reversed (the dense linear order itself is not ω-stable). My question is whether the first implication can be reversed. In other words, this is the same question as asking about the existence of [A totally categorical structure with trivial geometry which is not interpretable in the trivial structure](https://mathoverflow.net/questions/68748/a-totally-categorical-structure-with-trivial-geometry-which-is-not-interpretable).



ω-categorical, ω-stable structure with trivial geometry not definable in the pure set
The answer to my question was given to me by Ehud Hrushovski, and is as follows
(the following formulation is my own).


Let $D$ be the pure set.
For each pair $\{a,b\}$ of distinct elements of $D$ let $C\_{\{a,b\}}$ be a directed square, i.e., a directed cycle of length four, and let $f\_{\{a,b\}}$ map the two elements of one of the diagonals of $C\_{\{a,b\}}$ to $a$ and the remaining two elements to $b$.
Let $C$ be the disjoint union, over all pairs $\{a,b\}$, of the cycles $C\_{\{a,b\}}$; $C$ is equipped with the directed edge relation $E$.
Let $f:C\to D$ be the disjoint union of all the maps $f\_{\{a,b\}}$.


Now, the two-sorted structure $M=(D,C,E,f)$ is $\omega$-categorical, $\omega$-stable, has disintegrated strongly minimal sets, but does not interpret in the pure set. However, $M$ does interpret in $(\mathbb Q,\le)$ (this also proves $\omega$-categoricity of $M$). It is also a finite cover of a structure which interprets in the pure set, and a special case of a construction studied in Section 4 of the paper 'Totally categorical structures' by Hrushovski.


