Exercise 4, § 1

Let G be an ordered group, let P be the set of positive elements of G, and let f be the natural map from G onto a quotient group G/H. For f(P) to make G/H an ordered group it is necessary and sufficient that 0 ≤ y ≤ x and x ∈ H imply y ∈ H ; then H is called a convex subgroup of G, and G/H is considered as an ordered group in this way. If G is lattice ordered, then a necessary and sufficient condition for G/H to be lattice ordered and for $f(\sup(x, y)) = \sup(f(x), f(y))$ is that the relations |y| ≤ |x| and x ∈ H imply y ∈ H ; show that this says that H is a filtered convex subgroup. Show that a lattice ordered group with no filtered convex subgroups other than itself and {0} is totally ordered (consider filtered convex subgroups generated by two positive elements of G) ; \* deduce (Gen. Top., V, p. 25, Ex. 1) that G is then isomorphic to an additive subgroup of the real numbers. \*