Group with a compatible partial order
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 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 "a" ≤ "b" if and only if 0 ≤ -"a" + "b".
So we can reduce the partial order to a monadic property: "a" ≤ "b" if and only if -"a" + "b" ∈ "G"+.
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" · "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 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 "x"1, "x"2, "y"1, "y"2 are elements of "G" and "xi" ≤ "yj", then there exists "z" ∈ "G" such that "xi" ≤ "z" ≤ "yj".
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.
Properties.
Archimedean.
The Archimedean property of the real numbers can be generalized to partially ordered groups.
Property: A partially ordered group formula_1 is called Archimedean when for any formula_2, if formula_3 and formula_4 for all formula_5 then formula_6. Equivalently, when formula_7, then for any formula_8, there is some formula_9 such that formula_10.
Integrally closed.
A partially ordered group "G" is called integrally closed if for all elements "a" and "b" of "G", if "a""n" ≤ "b" for all natural "n" then "a" ≤ 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.
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.
Note.
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