I am working in an exercise from Professor K. Kunen's 2011 book. Let me write some definitions.
Given a relarion $R$, one defines the relation $R^*$ as follows:
$$aR^*b \leftrightarrow \exists n\in \omega \exists s \in A^{n+1} (a=s(0)R\dots Rs(n)=b).$$
It is clear that $R^*$ is the least transitive relation contaning $R$.
A relation $R$ is antitransitive iff:
$$\forall xyz(xRy\wedge yRz\rightarrow x\not R z)$$
The exercise is:
a) Prove that if $R$ is well founded defined on a set $A$ and every element has finite rank, there exists a unique antitransitive relation $H$ such that $H^*=R^*$.
b) Show that there is a countable well order that is conter example for this if we remove the finite rank hipothesis.
I managed to prove the existence of a) and showed that the uniqueness statement is in fact false: Let $A=4$, $R=<$, $H=\{(0, 1), (1, 2), (2, 3)\}$ and $\tilde H=\{(0, 1), (1, 2), (2, 3), (0, 3)\}$. Is the uniqueness really false or am I missing something?
For $b)$ I am having trouble finding a coumterexample. It looks like no ordinal up to $\omega.\omega+\omega$ works. Any help is appreciated.
Edit: exercise I.9.47.