In abstract algebra, a **partially ordered group** is a group *(G,+)* equipped with a partial order "≤" that is *translation-invariant*; in other words, "≤" has the property that, for all *a*, *b*, and *g* in *G*, if *a* ≤ *b* then *a+g* ≤ *b+g* and *g+a* ≤ *g+b*.

An element *x* of *G* is called **positive element** if 0 ≤ *x*. The set of elements 0 ≤ *x* is often denoted with *G*^{+}, and it is called the **positive cone of G**. So we have *a* ≤ *b* if and only if *-a*+*b* ∈ *G*^{+}.

By the definition, we can reduce the partial order to a monadic property: *a* ≤ *b* if and only if *0* ≤ *-a*+*b*.

For the general group *G*, the existence of a positive cone specifies an order on *G*. A group *G* is a partially ordered group if and only if there exists a subset *H* (which is *G*^{+}) of *G* such that:

*0* ∈ *H*
if *a* ∈ *H* and *b* ∈ *H* then *a+b* ∈ *H*
if *a* ∈ *H* then *-x*+*a*+*x* ∈ *H* for each *x* of *G*
if *a* ∈ *H* and *-a* ∈ *H* then *a=0*
A partially ordered group *G* with positive cone *G*^{+} is said to be **unperforated** if *n* · *g* ∈ *G*^{+} for some positive integer *n* implies *g* ∈ *G*^{+}. Being unperforated means there is no "gap" in the positive cone *G*^{+}.

If the order on the group is a linear order, then it is said to be a linearly ordered group. If the order on the group is a lattice order, i.e. any two elements have a least upper bound, then it is a **lattice-ordered group** (shortly **l-group**, though usually typeset with a script ell: ℓ-group).

A **Riesz group** is an unperforated partially ordered group with a property slightly weaker than being a lattice ordered group. Namely, a Riesz group satisfies the **Riesz interpolation property**: if *x*_{1}, *x*_{2}, *y*_{1}, *y*_{2} are elements of *G* and *x*_{i} ≤ *y*_{j}, then there exists *z* ∈ *G* such that *x*_{i} ≤ *z* ≤ *y*_{j}.

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

Partially ordered groups are used in the definition of valuations of fields.