Free lattice-ordered groups (Q5896513)

Let \(F_ r\) be the free lattice ordered group of finite rank \(r\geq 2\). A representation is obtained for \(F_ r\) as an \(\ell\)-subgroup of the \(\ell\)-group of all o-permutations of the real line. Using this representation the author proves the following. \(F_ r\) admits a right order Q such that \(P=\cap x^{-1}Qx\) for all \(x\in F_ r\) is the lattice order on \(F_ r\). Also P can be extended to a total two sided order on \(F_ r\). A free \(\ell\)-group has unique root extraction and posesses a solvable infrainvariant system of subgroups.