Exercise 12, § 1

Given a lattice ordered group G, a subgroup H of G is called a colattice if for each pair x, y of elements of H, $\sup_G(x, y) \in H$ (and so equals $\sup_H(x, y)$). a) If $G = \mathbf{Q} \times \mathbf{Q} \times \mathbf{Q}$ (where Q has the usual ordering), then the subgroup H of (x, y, z) such that z = x + y is a lattice but not a colattice. b) Every filtered convex (Ex. 4) subgroup H of a lattice ordered group G is a colattice. c) Let G be a lattice ordered group and H an arbitrary subgroup of G. Define H' to be the set of infima of finite subsets of H, and H" to be the set of suprema of finite subsets of H'. Show that H" is the smallest colattice of G containing H (use the remark on VI, p. 16).