Conjugacy in lattice-ordered groups and right ordered groups

Abstract

We prove that if G j are right orderable groups with non-trivial cyclic subgroups C j ( j = 1,2), then the (group) free product of G 1 and G 2 with C 1 and C 2 amalgamated is right orderable. We will deduce this from our main theorem that every lattice-ordered group can be embedded in one in which there are precisely four conjugacy classes. As a consequence of our method, we also show that every right ordered group can be embedded in a right ordered group in which any two non-identity elements are conjugate.