A **linearly ordered group** is a mathematical structure that combines the properties of a group with those of a linear order. More specifically, it is a group \( G \) equipped with a total order \( < \) that is compatible with the group operation.
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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.