Linearly ordered group

In mathematics, a linearly ordered group is both a group and a linearly ordered set, in which the group operation is in a certain sense compatible with the linear ordering. Specifically, we have

• For any x in the group G, either x ≥ 0 or −x ≥ 0, but not both, and
• For any x, y, z in G, if x ≤ y, then x + z ≤ y + z.

(See also ordered group.)

Otto Hölder showed that every linearly ordered group satisfying an Archimedean property is isomorphic to a subgroup of the additive group of real numbers.

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