I'm looking at a certain class of groups $G$ that come with a partial order $\le$ on the elements. So far it looks like $(G,\le)$ has the following properties:
The partial order is invariant under left translation and right translation.
The partial order is directed; in fact there is a cyclic subgroup that provides upper and lower bounds for any finite subset of the group.
There is a group homomorphism $I:G \rightarrow (\mathbb{Z},+)$ such that every positive element of $G$ is a product of positive elements that lie in $I^{-1}(\{1\})$.
Given $g \in G$, there is a canonical decomposition of $g$ as $g = g_pg_+g_-$, where $g_p,g_+,g_-$ commute with each other, $g_p$ has finite order (in the group theory sense, i.e. $g^n_p$ is the identity for some $n>0$); $g_+$ and $g^{-1}_-$ are positive; and given any $h \ge g$ such that $h$ is positive, then $h$ is also greater than or equal to the product of any subset of $\{g_p,g_+,g_-\}$.
It's more special than an arbitrary directed group, and property 4 looks a bit like what happens in a lattice-ordered group, except for the appearance of elements of finite order (of which there are many: the group contains a copy of every finite group as a subgroup). But at the same time, the elements of finite order mean that it can't be a lattice-ordered group, nor a quasi-lattice ordered group in the sense of Nica, nor can it have the Riesz interpolation property. (The typical situation is that a finite subset of $G$ has more than one minimal upper bound, and the minimal upper bounds differ from each other by elements of finite order.)
Has anyone studied a similar structure to this?
