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 $x\equiv y\iff x\leq y\leq x$. 

**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.

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 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.

I do know that every preorder $Q$, as a structure, is bi-interpretable with a certain partial order $P$, an order that in a sense codes the original preorder. One splits up the clusters of $Q$ into little antichains, and adds a maximal element above each one. Thus, in $P$ we can 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 not every partial order will arise as such a coding order, and so this idea does not provide a bi-interpretation of the theories. 

I strongly suspect that the theories are not bi-interpretable, but I've realized I don't know how to prove this.