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.

It is easy to see that these theories are mutually interpretable, but my question is specifically about bi-interpretation.

For clarity, one theory $S$ is interpretable in another theory $T$, if for some $n$ (the dimension) we can in $T$ define a class of $n$-tuples and an equivalence relation on them, and provide $T$-formulas to be used for the interpretations of the relations, functions, and constants of $S$, such that $T$ proves every resulting translation of an assertion in $S$, using the equivalence relation as the interpretation of $=$. A classic example is the interpretation of the rational field in the integer ring, using pairs (with second coordinate not zero) under the same-ratio equivalence relation.

Two theories are mutually interpretable, if each is interpreted in the other. Two theories are bi-interpretable, in contrast, a significantly stronger notion, if they are mutually interpretable in such a manner that each theory can also define a copy of its own universe in the interpreted copy of a copy. In terms of models, each model is interpreted in the other, and when doing it twice, each model is copied to the copy of itself within the copy of the other model. The critical requirement is that the resulting isomorphisms are definable in the parent model.

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.

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, each with its own private maximal element above, 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.

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 every partial order you can uniformly definably interpret a preorder, and in every preorder you can uniformly definably interpret a partial order, such that if you iterate this from order to preorder to order, the order you get is definably isomorphic with the original order, and similarly with the other iteration. In these interpretations, we allow the interpreted domain to consist of $k$-tuples modulo a definable congruence (like interpreting the complex field $\mathbb{C}$ in $\mathbb{R}$ or like the quotient field construction).

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, each with its own private maximal element above, 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.