Amalgamations of Lattice Ordered Groups

The author considers the problem of determining whether certain classes of lattice ordered groups (¿-groups) have the amalgamation property. It is shown that the classes of abelian totally ordered groups (ogroups) and abelian /-groups have the property, but that the class of ¿-groups does not. However, under certain cardinality restrictions one can find an ¿-group which is the "product" of /-groups with an amalgamated subgroup whenever (a) the ¿-subgroup is an Archimedian o-group, or (b) the ¿-subgroup is a direct product of Archimedian o-groups and the ¿-groups are representable. This yields a new proof that any /-group is embeddable in a divisible ¿-group, and implies that any ¿-group is embeddable in an ¿-group in which any two positive elements are conjugate.