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.

The class of orderable groups is closed under taking subgroups and direct products; it is locally closed, and consequently, a quasi-variety. A free product of orderable groups is again an orderable group.

References

[1]	A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)
[2]	L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)