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The main objective of this paper is to present several constructions of free products in the class of abelian /-groups which are sufficiently concrete to allow for an in depth examination of their structure. Some applications of these constructions are discussed, and it is shown that abelian /-group free products satisfy the subalgebra property. Further, some questions on free /-groups over group free products are considered for a variety of /-groups which is either abelian or contains the representable /-groups. Finally, a general observation is made about countable chains and countable disjoint sets in free algebras.1. Introduction. Let % be a class of /-groups (lattice ordered groups) and (G ι ,\ i G ί) a family of members of %. The %-free product of this family is an /-group G G %, denoted by % |J/e3 Gi> together with a family of /-monomorphisms (a ι ,:generates G as an /-group; (ii) for every H G % and every family of /-homomorphisms (β,: G, -> i/| / G ί), there exists a (necessarily) unique /-homomorphism β: G -> H satisfying β t = βa t for all / G ί. Following the usual practice we shall speak of ^LJ/es^ as the fyUfree product of (G, | i ε ί). To simplify our notation, we use the "internal" definition of a %-free product, that is, we identify each free factor G, with its image a^G;) in ^\J iG jG i9 and thus we think of each G, as an /-subgroup of ^UieίGj As a consequence of general existence theorems (See Gratzer [13, p. 186] or Pierce [25, p. 107]), tyUfree products always exist in any class of /-groups closed under products and /-subgroups.In this paper we concentrate on the class & of abelian /-groups, although many of our results also hold in the important class of vector lattices. Our main goal is to develop a reasonable representation theory for έE-free products. This is done in §2 where we give several methods of constructing these products, among the most useful of which represents β Uieί^ί (G/ e ί£) as a subdirect product of totally ordered abelian groups each determined by the primes of the individual G/s. We also show here how the Φ free products relate to the free abelian /-groups over partially ordered abelian groups.The third and fourth sections of the paper are devoted to considering several different properties for free products of /-groups. In particular using the representation theory established in §2 we show that the subalgebra property is satisfied for έMree products.

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This paper establishes an important link between the class of abelian (J-groups and the class of distributive lattices with a distinguished element. This is accomplished by describing the distributive lattice free product of a family of abelian H-groups as a naturally generated sublattice of their abelian f-group free product.

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