Exercise 29, § 1

An ordered group G is completely lattice ordered if it is filtered and if every nonempty subset of G which is bounded above has a supremum in G. A lower semi-lattice ordered monoid is said to be completely semi-Lattice ordered if $\inf(x_\alpha)$ exists for every nonempty family of elements $(x_i)$ of M which is bounded below, and if

for any two nonempty families $(x_i), (y_j)$ which are bounded below. A completely lattice ordered group is a completely semi-lattice ordered monoid. a) Show that a necessary and sufficient condition for a filtered group G to be completely lattice ordered is that every nonempty subset of the set P of positive elements of G admit an infimum in P. b) Every product of completely lattice ordered groups is completely lattice ordered. c) Every filtered convex subgroup of a completely lattice ordered group is completely lattice ordered.