Permutational Products of Lattice Ordered Groups

Let H be a group, let {G(: i e 1} be a set of groups and, for each i, let 9t be a a monomorphism: H -* Gt, with H9t = Hi. We call such a system of groups and monomorphisms an amalgam and denote it by [G{; H; 9,; / / , ] . By an embedding of the amalgam into a group G is meant a set of monomorphisms (p; : G; -*• G such that 9i<pi = 0j<Pj-, for all i,j and G,<P; n Gjcpj — H9k<pk, for all i,j, k. It is known (B. H. Neumann [9]) that the amalgam [Gt; H; 9t; Ht] can be embedded in a group. It is also known (J. M. Howie [7j) that if the Gt are just semigroups and the if, are almost unitary subsemigroups then the amalgam may be embedded in a semigroup. If G is an 1-group (lattice ordered group), a subgroup which is also a sublattice is called an 1-subgroup. If His an 1-subgroup of G and if the mapping C -» C n H is a one-to-one correspondence between the lattice of 1-subgroups of G and those of if then Conrad [2] called G an a-extension of H. In his discussion of a-extensions Conrad has shown that amalgams of the form [G;; H; 9t; Ht] are embeddable where the G; are a-extensions of the H{ and the Ht all belong to one of a number of classes of 1-groups. In a sense, the second section of this note considers the other extreme to Conrad's results. In section two, we consider the problem of embedding the amalgam [G,; H; 0,; Ht], where H and Gt are 1-groups, the Ht are convex 1-subgroups and the 9t are 1-monomorphisms (lattice and group monomorphisms), into an 1-group G. It is shown that if, for each i, Ht is in the centre of Gt then the amalgam is embeddable and if each Gt is abelian then the amalgam may be embedded in an abelian 1-group. The approach combines the permutational product of groups with the representation of an 1-group as an 1-subgroup of the group of automorphisms of an ordered set due to Holland [6]. If the G( are 0-groups (totally ordered groups) and the Hi are normal then the standard permutational product may be used. In general this does not yield an C-group although it will if the G; are abelian. In more general situations (e.g. [10]), consideration has been given to a weaker amalgamation embedding. Let us say that an amalgam [G;; H; 9t; Ht], where the