Linearly ordered group
= Linearly ordered group
{wiki}
= Totally ordered group
{synonym}
= o-group
{synonym}
A <group> with a two-sided invariant <total order>. Thus it is a <lattice-ordered group> whose <join> and <meet> choose the larger and smaller element. An order on only one side is not sufficient here. Every linearly ordered group is <torsion-free>: a positive element has strictly increasing positive powers, and a negative element has a positive inverse.