I need a hint to Exercise 8.13, Set Theory, Third Millennium Edition by Thomas Jech. The problem is to prove that
$ o\left({E_{\lambda_\alpha}^\kappa}\right) = \alpha $ . It seems to me that the essence is to prove that there can be no stationary set between $E_{\lambda_\alpha}^\kappa$ and $E_{\lambda_{\alpha+1}}^\kappa$.
In Chapter 8, Jech introduces the concept of stationary set and a well-ordering < on them. If $X$ and $Y$ are stationary sets of the cardinal $\kappa$, then $X<Y$ if the set of $\alpha\in Y$ such that $\alpha \cap X$ is NOT stationary in $\alpha$ is a non-stationary set.
Then, one defines $o\left( X \right)=\sup \{o\left( A \right)+1:A<X\}$
Finally, if $\kappa$ is a regular cardinal and $\lambda<\kappa$, then $E_\lambda^\kappa=\{\alpha<\kappa:cf\left(\alpha\right)=\lambda\}$ and $\{\lambda_\alpha:\alpha<\kappa\}$ is the increasing sequence of cardinals less than $\kappa$.