Partially ordered group

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 ab then a + gb + g and g + ag + b.

An element x of G is called positive if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G+, and is called the positive cone of G.

By translation invariance, we have ab if and only if 0 ≤ -a + b. So we can reduce the partial order to a monadic property: ab if and only if -a + bG+.

For the general group G, the existence of a positive cone specifies an order on G. A group G is a partially orderable 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 · gG+ for some positive integer n implies gG+. 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 l: 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 x1, x2, y1, y2 are elements of G and xiyj, then there exists zG such that xizyj.

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.

Archimedean property of the real numbers can be generalized to partially ordered groups.

A G is called integrally closed if for all elements a and b of G, if anb for all natural n then a ≤ 1.[1]

This property is somewhat stronger than the fact that a partially ordered group is Archimedean, though for a lattice-ordered group to be integrally closed and to be Archimedean is equivalent.[2] There is a theorem that every integrally closed directed group is already abelian. This has to do with the fact that a directed group is embeddable into a complete lattice-ordered group if and only if it is integrally closed.[1]

Everett, C. J.; Ulam, S. (1945). "On Ordered Groups". Transactions of the American Mathematical Society. 57 (2): 208–216. doi:. JSTOR .