A structure theorem and left-orderability of a quotient of quasi-isometry group of the real line

It is well-known that $QI(\mathbb{R})\cong (QI({\mathbb{R}}_{+})\times QI({\mathbb{R}}_{-}))⋊<t>$, where $QI(\mathbb{R})$(resp. $QI({\mathbb{R}}_{+})(\cong QI({\mathbb{R}}_{-}))$) is the group of quasi-isometries of the real line (resp. $[0,\mathrm{\infty })$). We introduce an invariant for the elements of $QI({\mathbb{R}}_{+})$ and split it into smaller units. We give an almost characterization of the elements of these units. We also show that a quotient of $QI({\mathbb{R}}_{+})$ gives an example of a left-orderable group which is not locally indicable.