partially ordered group

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.

Furthermore, every subgroup of G is a partially ordered group: it inherits the order from G.

The additive group of a partially ordered ring is always a partially ordered group.

A Riesz group is an unperforated partially ordered group with a property slightly weaker than being a lattice ordered group.

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.

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:

A partially ordered group G with positive cone G is said to be unperforated if n g G for some natural number n implies g G. Being unperforated means there is no "gap" in the positive cone G.

More generally, if G is a partially ordered group and X is some set, then the set of all functions from X to G is again a partially ordered group: all operations are performed componentwise.