Amalgamations of lattice ordered groups

Abstract

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