I am trying to find references that provide techniques for constructing (totally-) ordered subgroups of lattice-ordered groups $G$. For those who are not familiar, a lattice-ordered group $G$ is set which has both a group structure, as given by a binary operation $\cdot$, and a lattice structure, with ordering relation $\leq$, meet $\land$, and join $\lor$, such that $g_1\leq g_2$ implies that $x\cdot g_1\leq x\cdot g_2$ and $g_1\cdot x\leq g_2\cdot x$ for any $g_1,g_2,x\in G$.
All of the books/papers I've found online focus on convex subgroups of lattice-ordered groups, i.e. subgroups $S\subset G$ with the property that $a\leq x\leq b\implies x\in S$ for all $a,b\in S$ and $x\in G$. However, I am interested in totally ordered subgroups, in particular, whether there are any ways of canonically constructing maximal totally-ordered subgroups given certain properties of the group $G$. The existence of such subgroups can be proven easily using Zorn's Lemma, I think - but I'm looking for explicit constructions that don't involve arbitrary choices, especially ones in which the elements of the subgroup can be described explicitly by some simple property.
Reference request: Does anyone know of any references describing explicit techniques of constructing maximal totally-ordered subgroups of lattice-ordered groups?
Context: I'm working on an undergraduate thesis that involves the algebraic structure of asymptotic growth orders of sequences of positive real numbers, and have figured out that a certain subset of these growth orders can be endowed with the structure of a lattice-ordered group. One of my goals is to identify a subset of the set of all growth orders which is totally-ordered but also "big" (at least, big enough to be closed under certain operations like multiplication or composition), so that any two growth orders in the subset are comparable, as a way of eliminating "undesirable" growth orders with oscillatory behavior. My hope is that the lattice-ordered group structure will help me find such a chain. (Also, here is a link to the current draft of my thesis.)