Linearly ordered group

In terms of partially ordered group
A linearly ordered group is a partially ordered group satisfying the following equivalent conditions:


 * 1) The underlying partial order is a total order (linear order), i.e., any two elements of the group are comparable.
 * 2) For every element of the group, either that element or its inverse is in the positive cone of the group (note that the identity element is the only element such that both that and its inverse are in the positive cone).

Facts

 * Abelian implies linearly orderable iff torsion-free