I am trying to read the chapter on PCF theory by Abraham and Magidor in Handbook of Set Theory. The authors first consider structures of form $(P,<_{P},\leq_{P})$ where $<_{P}$ is irreflexive and transitive, while $\leq_{P}$ is reflexive and transitive. Then they define three notions of upper bound, and claim that (page 1155):
(a) In the case of ordinal functions $\mathrm{On}^{A}$ from a fixed set $A$, together with a proper ideal $I$ on $A$, the notions of least upper bound and minimal upper bound coincide.
(b) If $h$ is an upper bound of $F\subseteq\mathrm{On}^{A}$ and $F$ is cofinal in $\prod h/I$, then $h$ is actually a least upper bound.
I can not prove any of these. I can see that (a) is true when $I$ is a prime ideal but clearly that is not assumed here. Why is it impossible to have incomparable minimal upper bounds? For (b), cofinal means for any $g<_{I} h$ there exists $f\in F$ such that $g<_I f$; why is it impossible to have another upper bound $k$ such that $h \nleq_{I} k$, namely $k(a)<h(a)$ for a "positive measure of $a$" (this does not imply $k<_{I}h$)?
Also, the chapter in Handbook seems to be the most friendly introductory text (much better than Jech) and yet I constantly get stuck. Is there any other suggestion?