Exercise 8, § 2

An ordered set E is said to be ramified (on the right) if, for each pair of elements $x$, $y$ of E such that $x < y$, there exists $z > x$ such that $y$ and $z$ are not comparable. E is said to be completely ramified (on the right) if it is ramified and has no maximal elements. Every antidirected set (§ 1, Exercise 22) is ramified.

(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 (§ 1, Exercise 24), show that every maximal element of $\mathfrak{R}_\alpha$ is completely ramified.

(c) Give an example of a branched set which is not ramified. The branched set defined in §1, Exercise 24 (c) is completely ramified.

(d) Let E be a set in which each interval $]\leftarrow, x]$ is totally ordered. Show that E has an antidirected cofinal subset (§1, Exercise 22) (use (b)).