This is a followup question related to this question. Recall that a left-invariant partial order on a finitely generated group $G$ is called a partial *word order* if for every $a\le b\le c$ we have $|b|\le C(|a|+|c|)$ for some constant $C$ (where $|x|$ is the word length of $x$). For example, the following partial order on ${\mathbb Z}^2$: $(m,n)\le (k,l)$ iff $m\le k, n\le l$ is a word partial order. Certainly the empty (trivial) partial order ($a\le b$ iff $a=b$) is always a word order.

** Question 1. ** Is there a canonical way to construct non-trivial partial word orders on groups?

** Update 2** A sub-question: is there a general algebraic property of a group that guaranties existence of such non-trivial partial orders? (Being free Abelian of finite rank is such a property, but I am looking for non-trivial answers.)

** Update 1** *One possible generalization.* I call a partial order on a group *quasi-invariant* if for every $g,a,b$ if $a\lt b$, then there exist two elements $c,d$ such that
$gac\lt gbd$ and $|c|,|d|\lt C$ for some constant $C$.

** Question 2. ** Did anybody study quasi-invariant partial orders on groups?

** Motivation** The reason I want to study such things is to introduce an extra structure on the asymptotic cones of groups. If one tries to define a partial order on an asymptotic cone, one would need a quasi-(left)-invariant word partial order on the group.