I must first preface that while this is indeed a question on an exercise, I believe this is advanced enough for MathOverflow.


Let $\kappa$ be a regular uncountable cardinal. Recall that the notion of a stationary subset makes sense for subsets of limit ordinals of uncountable cofinality. In Jech's *Set Theory* (Third Millenium Edition), he defined an ordering $<$ on stationary subsets $S,T \subseteq \kappa$ by (Definition 8.18, or (2.6) of the linked paper below):
$$
S < T \iff S \cap \alpha \text{ is stationary for almost all } \alpha \in T
$$
Note that we implicitly assume that almost all $\alpha \in T$ are limit ordinals of uncountable cofinality. The standard examples of such stationary sets are of the form:
$$
E\_\lambda^\kappa := \{\alpha < \kappa : \operatorname{cf}(\alpha) = \lambda\}
$$
This was first introduced in his paper [Stationary subsets of inaccessible cardinals](https://drive.google.com/file/d/1VdcZ17N5uh7Vo4FnbkvQI-OiZf_lXq9n/view?usp=sharing). He proved in Lemma 8.19 (Theorem 2.4 of the paper) that $<$ is a well-founded relation, so it makes sense to define a rank function on this relation, $o(S)$, which he calls the *order* of the set.


He then proceeds to give two exercises on this matter:


1. (Exercise 8.13) If $\lambda < \kappa$ is the $\alpha^\text{th}$ regular cardinal, then $o(E\_\lambda^\kappa) = \alpha$.
2. (Exercise 8.14) $o(\kappa) \geq \kappa$ if and only if $\kappa$ is weakly inaccessible; $o(\kappa) \geq \kappa + 1$ if and only if $\kappa$ is weakly Mahlo.


I have no idea how to solve either exercise. It may be helpful to note that in his paper, he defined the notion of a *canonical stationary set (of order $\nu$)*, and mentioned without proof that the set $E\_\lambda^\kappa$ is the canonical stationary set of order $\lambda$. However, I do not see why this is true.



Properties of Jech's hierarchy of stationary sets (Exercise 8.13, 8.14 of Jech)
After some googling I found [the notes by J. D. Monk](http://euclid.colorado.edu/%7Emonkd/jech.pdf), which have answered the questions in the span of Theorem 2.68 to 2.91.


