# Ordered group

In abstract algebra, an **ordered group** 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*.
Note that sometimes the term *ordered group* is used for linearly/totally ordered group, and what we describe here is called *partially ordered group*.

By the definition we can reduce the partial order to a monadic property: *a* ≤ *b* iff *1* ≤ *a*^{-1} *b*. The set of elements *x* ≥ *1* of *G* are often denoted with *G*^{+}. So we have *a* ≤ *b* iff *a*^{-1}*b* ∈ *G*^{+}. That *G* is an ordered group can be expressed only in terms of *G*^{+}: A group is an ordered group iff there exists a subset *H* (which is *G*^{+}) of *G* such that:

*1*∈*H**a*∈*H*and*b*∈*H*then*ab*∈*H*- if
*a*∈*H*then*x*^{-1}*ax*∈*H*for each*x*of*G*

A typical example of an ordered group is **Z**^{n}, where the group operation is componentwise addition, and we write (*a*_{1},...,*a*_{n}) ≤ (*b*_{1},...,*b*_{n}) iff *a*_{i} ≤ *b*_{i} (in the usual order of integers) for all *i*=1,...,*n*.
More generally, if *G* is an ordered group and *X* is some set, then the set of all functions from *X* to *G* is again an ordered group: all operations are performed componentwise. Furthermore, every subgroup of *G* is an ordered group: it inherits the order from *G*.

If the order on the group is a linear order, we speak of a linearly ordered group.

If *G* and *H* are two ordered groups, a map from *G* to *H* is a *morphism of ordered groups* if it is both a group homomorphism and a monotonic function. The ordered groups, together with this notion of morphism, form a category.

Ordered groups are used in the definition of valuations of fields.