Exercise 31, § 1

An ordered group G is called archimedean if the only elements $x \in G$ such that the set of $nx$ (n a positive integer) is bounded below are the positive elements of G.

a) The monoid $\mathscr{M}(G)$ of major sets in an ordered group G is a group if and only if G is archimedean (use Ex. 30, g) and d)). If in addition G is filtered, then $\mathscr{M}(G)$ is a completely lattice ordered group. b) Deduce that a filtered group G is isomorphic to a subgroup of a completely lattice ordered group if and only if G is archimedean.