An ordered group is both a poset and a group in a compatible way. The concept applies directly to other constructs with group structure, such as ordered abelian groups, ordered vector spaces, etc. However, for ordered ring?s, ordered fields, and so on, additional compatibility conditions are required.
Let be a group (written additively but not necessarily commutative), and let be a partial order on the underlying set of . Then is an ordered group if this compatibility condition (translation invariance) holds:
If is an abelian group, then we have an ordered abelian group; in this case, only one direction of translation invariance needs to be checked.
The order is determined entirely by the group and the positive cone :
It's possible to define an ordered group in terms of the positive cone (by specifying precisely the conditions that the positive cone must satisfy); see positive cone for this.
However, this characterisation probably can't be made to work for ordered monoids (although I haven't checked for certain).
The underlying additive group of any ordered field is an ordered group.
In particular, the underlying additive group of the field of real numbers is an ordered group.
Although the field of complex numbers is not an ordered field (since it is not linearly ordered), its underlying additive group is still an ordered group (where means that is a nonnegative real number).
Given a topological vector space , we often consider its dual vector space , consisting of the continuous linear maps from to its base field, which is usually either or . This inherits a partial order from the target field, and then the underlying additive group is an ordered group; in fact, we have an ordered algebra?. (This is the main sort of example that I know of, but that probably just reflects my own limited knowledge.)