A partial order relation $\leq$ on a set $A$ is a binary relation that is reflexive, transitive, and antisymmetric.

A preorder relation $\unlhd$ (also sometimes known as a quasi order or pseudo order) is merely reflexive and transitive. The difference is that a preorder may have nontrivial clusters of equivalent nodes, equivalent via $x\equiv y\iff x\unlhd y\unlhd x$.

Question. Is the theory of a partial order bi-interpretable with the theory of a preorder?

It is easy to see that these theories are mutually interpretable, but my question is specifically about bi-interpretation. I had intended to use these theories as an elementary example in the book I am writing in the section on interpretations of models and theories, but I have become confused about whether they are bi-interpretable or not. I strongly suspect that they are not bi-interpretable.

For the theories to be bi-interpretable would mean that in any partial order you can uniformly define a preorder on a domain of $k$-tuples modulo a definable congruence relation, and similarly in any preorder you can define a partial order on a domain of $k'$ tuples modulo a definable congruence, such that for any partial order $P$, if you interpret the preorder $Q=P^k/\simeq$, and then inside $Q$ interpret the partial order $P'=Q^{k'}/\approx$, then $P$ is isomorphic to $P'$ by a map that is definable in $P$; and similarly, if you start with a preorder $Q$ and interpret the partial order $P$ and then the preorder $Q'$ inside that, then $Q$ is definably isomorphic to $Q'$.

In particular, if the theories were bi-interpretable, then this would provide a bijective correspondence between isomorphism types of partial orders and preorders in such a way that is definable and interpretable, and every order could see for itself how it is copied through the iterated interpretations.

Of course every preorder $Q$ has the natural quotient $Q/\equiv$, which is a partial order interpretable in $Q$. But this construction will not help with a bi-interpretation, since many different preorders have isomorphic quotient orders — you cannot recover the original preorder from the quotient order.

Meanwhile, I do know that every preorder $Q$, as a structure, is bi-interpretable with a certain partial order $P$, an order that codes the original preorder in a tighter manner. Namely, one splits up the clusters of $Q$ into little antichains, and adds a maximal element above each one, thereby marking it as such. In the resulting order $P$, we can definably identify which are the original nodes, and in $Q$ we can simulate $P$ and in $P$ we can simulate $Q$ in a structure-bi-interpretation manner.

But this is not a bi-interpretation of the theories, since not every partial order will arise as such a coding order. I wonder if one can somehow fix things up so that every order codes a preorder and conversely? Actually, I think it is not true.