Orderable group

A group $G$ on which a total order $\leq$( cf. Totally ordered group) can be introduced such that $a \leq b$ entails $x ay \leq x b y$ for any $a , b , x , y \in G$. A group $G$ is orderable as a group if and only if there exists a subset $P$ with the properties: 1) $P P \subseteq P$; 2) $P \cap P ^ {-} 1 = \{ 1 \}$, 3) $P \cup P ^ {-} 1 = G$; 4) $x ^ {-} 1 P x \subseteq P$ for any $x \in G$.
Let $S ( a _ {1} \dots a _ {n} )$ be the normal sub-semi-group of $G$ generated by $a _ {1} \dots a _ {n}$. The group $G$ is orderable as a group if and only if for any finite set of elements $a _ {1} \dots a _ {n}$ in $G$, different from the unit element, numbers $\epsilon _ {1} \dots \epsilon _ {n}$ can be found, equal to $\pm 1$, such that the sub-semi-group $S ( a _ {1} ^ {\epsilon _ {1} } \dots a _ {n} ^ {\epsilon _ {n} } )$ does not contain the unit element. Every orderable group is a group with unique root extraction. Abelian torsion-free groups, locally nilpotent torsion-free groups, free, and free solvable groups are orderable groups. Two-step solvable groups such that for every non-unit element $x$, $1 \notin S ( x)$, are orderable groups.