Exercise 15, § 1

(a) Let E be an ordered set, and for each subset X of E let $\rho(\mathrm{X})$ (resp. $\sigma(\mathrm{X})$) denote the set of upper (resp. lower) bounds of X in E. Show that, in $\mathfrak{P}(\mathrm{E})$, the set $\tilde{\mathrm{E}}$ of subsets X such that $\mathrm{X} = \sigma(\rho(\mathrm{X}))$ is a complete lattice, and that the mapping $i : x \to \sigma(\{x\})$ is an isomorphism (called canonical) of E onto an ordered subset E' of $\tilde{\mathrm{E}}$ such that, if a family $(x_\iota)$ of elements of E has a least upper bound (resp. greatest lower bound) in E, the image of this least upper bound (resp. greatest lower bound) is the least upper bound (resp. greatest lower bound) in $\tilde{\mathrm{E}}$ of the family of images of the $x_\iota$. $\tilde{\mathrm{E}}$ is called the completion of the ordered set E.

(b) Show that, for every subset X of E, $\sigma(\rho(\mathrm{X}))$ is the least upper bound in $\tilde{\mathrm{E}}$ of the subset $i(\mathrm{X})$ of $\tilde{\mathrm{E}}$. If $f$ is any increasing mapping of E into a complete lattice F, there exists a unique increasing mapping $\bar{f}$ of $\tilde{\mathrm{E}}$ into F such that $f = \bar{f} \circ i$ and $\bar{f}(\sup \mathrm{Z}) = \sup (\bar{f}(\mathrm{Z}))$ for every subset Z of $\tilde{\mathrm{E}}$.

(c) If E is totally ordered, show that $\tilde{\mathrm{E}}$ is totally ordered.