I'm trying to do the problem 7.17 on th book Notes of Set Theory of Moschovakis.
First I will define what is a prewellordering:
A prewellordering on a set $A$ is any relation $(\lesssim) ⊆ A×A$ which is reflexive, transitive, connected (total) and grounded. “Connected” means that any two points in $A$ are comparable, $$(\forall x, y \in A)[x y \vee y x],$$ and “grounded” means that every non-empty $X ⊆ A$ has a $\lesssim$ -least member, $$(\forall X ⊆ A,X = \emptyset)(\exists x \in X)(∀y \in X)[x y].$$ A prewellordering would be a wellordering, if only it were antisymmetric.
I have to prove that
A relation $\lesssim ⊆ A \times A $ is a prewellordering if and only if there exists a well ordered set $(U,≤_{U})$ and a surjection $\pi: A \longrightarrow U$ such that $$x \lesssim y > \Longleftrightarrow \pi(x) ≤_{U} \pi(y) (x, y ∈ A).$$
So, here is my approach. I think that if le relation is a prewellordering maybe we can construct a trivial surjection such as $\pi(x)=\emptyset \forall x$. The set $\{\emptyset\}$ is well ordering right?
On the other hand, on the other sense, I think is easy to prove that $\lesssim$ is reflexive, transitive by doind a direct proof and using $\leq_U$ is a well order. Then, the last thing is showing that any subset $X\subset A$ has a $\lesssim$-minimal element. We can take $\pi(X)$, its $\leq_U$-minimal element, $m$, and so the minimal element on $X$ would be $\pi^{-1}(m)$.
Do you think I've maken any mistake?
Yes, you've made a mistake. If you map everything to $0$, and there were $x,y$ such that $x\lesssim y$ but $y\not\lesssim x$, then it can't be that $\pi(x)=\pi(y)=0$.
The idea is that if you have a preorder, then there is a natural quotient which gives a partial order. And a prewellordering is such where the quotient is a well-ordering.
And the key point is that you need to solve the problem of anti-symmetry. How do you do that? By noting that if you consider the equivalence relation: $x\sim y\iff x\lesssim y\land y\lesssim x$, then the preorder induces a partial order on the equivalence classes.
I will leave it to you to see how to use this to finish the exercise.