I am confused about something. If (all what I will write are operators) $x$ is compatible with $p_y$ that means they have the same eigenvectors. However, $x$ is compatible with $y$ which means they have the same eigenvectors. That makes the eigenvectors of $p_y$ and $y$ are the same which makes them compatible. But it's not the case.
Where is the logic wrong?
$x$ is compatible with $p_y$ that means they have the same eigenvectors.
That is perhaps not the best way to say it. The fact that $x$ and $p_y$ are compatible tells you that there exists at least one set of basis vectors which are simultaneously eigenvectors of both $x$ and $p_y$. Let's call this set $\{v\}$. Similarly, the fact that $x$ and $y$ are compatible means that there exists at least one set of basis vectors which are simultaneously eigenvectos of both $x$ and $y$. Let's call this set $\{w\}$. Nothing guarantees that $\{v\}$ and $\{w\}$ are the same set. Therefore, the statement that $y$ and $p_y$ are compatible is not necessarily true, and is as you know in fact not true.
