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.
In abstract algebra, an orderedgroup is a group G equipped with a partial order "≤" which is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then ag ≤ bg and ga ≤ gb.
If G and H are two orderedgroups, a map from G to H is a morphism of orderedgroups if it is both a group homomorphism and a monotonic function.
Orderedgroups are used in the definition of valuations of fields.
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