Exercise 32, § 1

a) Let G be a completely lattice ordered group, and H a subgroup of G. For every major subset A of the group H, let $x_A$ denote the infimum of A in G; show that the map $A \mapsto x_A$ from $\mathscr{M}(H)$ to G is injective (show that A is the set of $y \in H$ such that $y \geq x_A$). b) For every subset B of H, bounded below in H, let (B) denote the major set generated by B in H (an element of $\mathscr{M}(H)$). If for any subset B of H, bounded below in H, the equation $\inf B = \inf (B)$ holds (infima taken in G), show that the map $A \mapsto x_A$ is an isomorphism of the ordered group $\mathscr{M}(H)$ onto a subgroup of G (cf. Ex. 30, e)). c) If every element of G is the infimum of some subset of H, show that for any subset B of H, bounded below in H, we have $\inf B = \inf (B)$ (infima taken in G). (Notice that this holds if $\inf B \in H$; in general, consider an element $x \in G$ such that $x + \inf B \in H$ and use the fact that $x$ is the infimum of some subset of G, as well as Ex. 30, e.). d) \* Let G be the completely lattice ordered group $\mathbf{Z} \times \mathbf{Z} \times \mathbf{R}$. Let $0 > 0$ be an irrational number; let H be the subgroup of $(x, y, z)$ such that $\theta(z - x) + y = 0$. Show that there exists no isomorphism of the ordered group $\mathscr{M}(H)$ onto any subgroup of G which restricts to the identity map on H (notice that $\mathscr{M}(H)$ is isomorphic to $K = \mathbf{Z} \times \mathbf{R}$; consider the subgroup of K consisting of elements $u$ such that, for every integer $n > 0$, there exists $v \in K$ with $nv = u$, and the analogous subgroup of G).