Exercise 30, § 6

Let E be a lattice such that the set obtained by endowing E with the opposite ordering is Noetherian. Show that every element $a \in E$ can be written as sup $(e_1, e_2, \ldots, e_n)$, where $e_1, \ldots, e_n$ are irreducible (§ 4, Exercise 7; show first that there exists an irreducible element $e$ such that $a = \sup (e, b)$ if $a$ is not irreducible). Generalize Exercise 7(b) of § 4 to E; also generalise Exercises 8(b) and 9(b) of § 4.