Exercise 29, § 6

(a) In an ordered set E, every finite union of Noetherian subsets (with respect to the induced ordering) is Noetherian.

(b) An ordered set E is Noetherian if and only if for each $a \in \mathrm{E}$ the interval $]a, \rightarrow[$ is Noetherian.

(c) Let E be an ordered set such that the ordered set obtained by endowing E with the opposite ordering is Noetherian. Let $u$ be a letter and let $\mathrm{T}\{u\}$ be a term. Show that there exists a set U and a mapping $f$ of E onto U such that for each $x \in \mathrm{E}$ we have $f(x) = \mathrm{T}\{f^{(x)}\}$, where $f^{(x)}$ denotes the mapping of $]\leftarrow, x[$ onto $f(]\leftarrow, x[)$ which coincides with $f$ on this interval. Furthermore, U and $f$ are determined uniquely by this condition.

(d) Let E be a Noetherian ordered set such that every finite subset of E has a least upper bound in E. Show that, if E has a least element, then E is a complete lattice (§ 1, Exercise 11); and that if E has no least element, the set $E'$ obtained by adjoining a least element to E (§ 1, no. 7, Proposition 3) is a complete lattice.