Free products of lattice ordered groups

The distributive lattice free product as a sublattice of the abelianl-group free product

“…This is accomplished with the use of Theorem 2.4 and the construction of the (2-free product given in Powell and Tsinakis (1981). (See Theorem 3.2 below.…”

“…This representation is a special case of a general representation theory for (£-free products (see Powell and Tsinakis (1981)). In addition to this paper and Franchello's work (1978), other contributions to the study of free products of C-groups have been made by Holland and Scrimger (1972), and Martinez (1972Martinez ( ), (1973.…”

Strong amalgamations of lattice ordered groups and modules

“…In [15] Powell and Tsinakis extended this result to all representable varieties containing one of the two solvable, non-nilpotent covers of A. Further, in [12] they showed that the varieties of nilpotent /-groups do not satisfy AP. To date no general proof has surfaced to show that AP fails in all nonabehan varieties of/-groups although this result is likely to be true.…”

“…For basic information on /-group free products and amalgamations, see Powell and Tsinakis [12], [16], and [17]. Background on lattice ordered groups and modules in general can be found in Bigard, Keimel and Wolfenstein [2].…”

“…Implicit in his work is a proof that the varieties above and including the non-representable covers of A also fail AP. Subsequently, Powell and Tsinakis showed in [12] and [13] that there exists an uncountable chain of varieties containing R and faihng AP such that their join is the largest proper variety of/-groups. It was later proved by Glass, Saracino, and Wood [4] that the variety R itself and many varieties contained therein cannot have the amalgamation property.…”