Right orders and amalgamation for lattice-ordered groups

Let H _i be a sublattice subgroup of a lattice-ordered group G _i (i = 1, 2). Suppose that H ₁ and H ₂ are isomorphic as lattice-ordered groups, say by φ. In general, there is no lattice-ordered group in which G ₁ and G ₂ can be embedded (as lattice-ordered groups) so that the embeddings agree on the images of H ₁ and H ₁ ^φ . In this article we prove that the group free product of G ₁ and G ₂ amalgamating H ₁ and H ₁ ^φ is right orderable and so embeddable (as a group) in a lattice-orderable group. To obtain this, we use our necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable BLUDOV, V. V.—GLASS, A. M. W.: Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup, Proc. London Math. Soc. (3) 99 (2009), 585–608]. We also provide new limiting examples to show that amalgamation can fail in the category of lattice-ordered groups even when the amalgamating sublattice subgroups are convex and normal (?-ideals) and solve of Problem 1.42 from KOPYTOV, V. M.—MEDVEDEV, N. YA.: Ordered groups. In: Selected Problems in Algebra. Collection of Works Dedicated to the Memory of N. Ya. Medvedev, Altaii State University, Barnaul, 2007, pp. 15–112 (Russian)].