Free Products of Lattice-Ordered Groups

Abstract

ABSTRACT. We give a highly homogeneous representation for the free product of two nontrivial countable lattice-ordered groups and obtain, as a consequence of the method, that the free product of nontrivial lattice-ordered groups is directly indecomposable and has trivial center.1. Introduction. The free lattice-ordered group on a countably infinite set of generators was shown to be isomorphic to a doubly transitive sublattice subgroup of the lattice-ordered group of all order-preserving permutations of the rational line (see [1 or 2, Theorem 6.7, and 9]). The proof is easily extended to give a faithful doubly transitive representation for the free lattice-ordered group on any infinite set of generators. More recently, Kopytov [6] and, independently, McCleary [9] proved by a far more subtle technique that the free lattice-ordered group on at least two generators possesses a faithful doubly transitive representation (on the rational line if the number of generators is finite or countable). The ideas were further extended in [3] to prove:1. If G is any lattice-ordered group and F is a free lattice-ordered group on an infinite set of generators of cardinality at least \G\, then the free product (in the category of lattice-ordered groups) of G and F has a faithful doubly transitive representation on some ordered field of cardinality \F\.2. If G is any countable lattice-ordered group and F is a free lattice-ordered group on a finite number (> 1) of generators, then the free product (in the category of lattice-ordered groups) of G and F has a faithful doubly transitive representation on the rational line.Although the proof of 2 is the more delicate, the purpose of this note is to extend it. (In contrast, I have been unable to extend the proof of 1.) Specifically,