Are homomorphic pre-images of partial orders preorders?

Question

Suppose we have a relation $R$ and a function $f$, along with a partial order $P$, such that if $f(x) P f(y)$, then $xRy$. It is easy to see that $R$ must then be a preorder (i.e. a reflexive and transitive relation). My question is, do all preorders arise in this way? In other words, is the class of homomorphic preimages of partial orders equal to the class of preorders?

Answer 1

What you're calling a "homomorphic pre-image" here is not what I would call a homomorphic pre-image. Does your definition come from some source, or did you make it up?

I would expect a homomorphic pre-image of a structure $B$ to just be a structure $A$ that admits a homomorphism to $B$. Unpacking the definition in the case of a single binary relation $R$: a structure $A$ is a homomorphic pre-image of a structure $B$ just when there is a function $f\colon A\to B$ such that for all $x,y\in A$, if $xRy$, then $f(x)Rf(y)$.

Eric Wofsey has already pointed out in the comments that under your definition, not every homomorphic pre-image of a partial order is a preorder (transitivity may fail). Under my defininition, it's also the case that not every homomorphic pre-image of a partial order is a preorder (e.g., reflexivity may fail: the empty relation is always a homomorphic preimage).

What is true is that every preorder $(X,\leq)$ admits a homomorphism to a partial order. Define an equivalence relation on $X$ by $x\sim y$ iff $x\leq y$ and $y\leq x$. Then the induced relation on $X/{\sim}$ is a partial order.