# In first-order logic without equality, is the theory of partial orders the same as preorders?

This is related to my previous question on equality and equivalence relations. In first-order logic without equality with a single binary relation $$R$$, is the theory of reflexive partial orders the same as the theory of preorders?

Suppose $$\mathfrak{A},\mathfrak{B}$$ are structures in the same relational language and $$R\subseteq\mathfrak{A}\times\mathfrak{B}$$ is a total surjective relation which preserves and reflects all the relations in that language. Then $$\mathfrak{A}\equiv_{\mathsf{FOLw/o=}}\mathfrak{B}$$; in fact, $$\mathfrak{A}$$ and $$\mathfrak{B}$$ have the same equality-free $$\mathcal{L}_{\infty,\infty}$$-theories.
This lets us prove results about classes of structures. Specifically, say that $$\mathfrak{A}$$ and $$\mathfrak{B}$$ are quasi-isomorphic if there is some $$R$$ with the properties above. Then we have:
Suppose $$\mathbb{K}_0\subseteq\mathbb{K}_1$$ are classes of structures in the same relational language, and every structure in $$\mathbb{K}_1$$ is quasi-isomorphic to a structure in $$\mathbb{K}_0$$. Then $$\mathbb{K}_1$$ and $$\mathbb{K}_0$$ have the same equality-free-first-order theories: the equality-free-first-order sentences true in every structure in $$\mathbb{K}_0$$ are also true in every structure in $$\mathbb{K}_1$$ (since $$\mathbb{K}_0\subseteq\mathbb{K}_1$$ the other direction is trivial).
In this case, take $$\mathbb{K}_1$$ to be the class of preorders and $$\mathbb{K}_0$$ to be the class of partial orders (with "partialization" giving the desired $$R$$s).