This is Exercise III.2.8 of Bourbaki's Theory of Sets.
An ordered set $E$ is said to be ramified if, for each pair of elements $x,y$ of $E$ such that $x
(a) Let $E$ be an ordered set and let $a$ be an element of $E$. Let $\mathfrak{R}_a$ denote the set of ramified subsets of $E$ which have $a$ as least element. Show that $\mathfrak{R}_a$, ordered by inclusion, has a maximal element.
(b) If $E$ is branched, show that every maximal element of $\mathfrak{R}_a$ is completely ramified.
(An ordered set $E$ is said to be branched if for each $x\in E$ there exist $y,z$ in $E$ such that $x\leq y$, $x\leq z$ and the intervals $\left[y,\rightarrow\right[$ and $\left[z,\rightarrow\right[$ do not intersect.)
The proof of (a) is a straightforward application of Zorn's Lemma.
For (b), I would like to argue in the following way. Suppose $F$ is a maximal element of $\mathfrak{R}_a$ and $x$ is a maximal element of $F$. Then there is $y,z$ in $E$ such that $x\leq y$, $x\leq z$ and $\left[y,\rightarrow\right[\cap\left[z,\rightarrow\right[=\emptyset$. Then $F\cup\{y,z\}$ is ramified.
But I haven't been able to show that $F\cup\{y,z\}$ is ramified, and I'm not sure that it's true, since if $b\in F$ and $b
Any hints at the solution or general thoughts about the problem are greatly appreciated.
Bourbaki seems to be the only mathematician to use the words "branched" and "ramified" for those properties. Are there other more common words, or are these properties without interest?