Exercise 36, § 1

Let G be a filtered group with positive elements P. Then G is isomorphic to a group of the form $\mathbf{Z}^{(1)}$ if and only if there exists a map $x \mapsto d(x)$ from P into N such that, for all b, either $b \geq a$ or there exists an element c in the major set (a, b) such that $d(c) < d(a)$. (Show that this condition is satisfied in $\mathbf{Z}^{(1)}$ by the map $d(x_i) = \sum x_i$ (when $x_i > 0$ for all i).

Conversely, if the condition is satisfied, show that every major set $A \subset P$ has the form (a), by taking an element a of A such that $d(a) \leq d(x)$ for all $x \in A$. Apply Th. 2 of VI, p. 18.)