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 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:

• 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 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 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.

• The integers with their usual order
• An ordered vector space is a partially ordered group
• A Riesz space is a lattice-ordered group
• A typical example of a partially ordered group is Zⁿ, where the group operation is componentwise addition, and we write (a₁,...,a_n) ≤ (b₁,...,b_n) if and only if a_i ≤ b_i (in the usual order of integers) for all i = 1,..., n.
• 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. Furthermore, every subgroup of G is a partially ordered group: it inherits the order from G.
• If A is an approximately finite-dimensional C*-algebra, or more generally, if A is a stably finite unital C*-algebra, then K₀(A) is a partially ordered abelian group. (Elliott, 1976)

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

Property: A partially ordered group ${\displaystyle G}$ is called Archimedean when for any ${\displaystyle a,b\in G}$, if ${\displaystyle e\leq a\leq b}$ and ${\displaystyle a^{n}\leq b}$ for all ${\displaystyle n\geq 1}$ then ${\displaystyle a=e}$. Equivalently, when ${\displaystyle a\neq e}$, then for any ${\displaystyle b\in G}$, there is some ${\displaystyle n\in \mathbb {Z} }$ such that ${\displaystyle b<a^{n}}$.

A partially ordered group G is called integrally closed if for all elements a and b of G, if aⁿ ≤ b 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]

• Cyclically ordered group – Group with a cyclic order respected by the group operation
• Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb
• Ordered field – Algebraic object with an ordered structure
• Ordered ring – ring with a compatible total orderPages displaying wikidata descriptions as a fallback
• Ordered topological vector space
• Ordered vector space – Vector space with a partial order
• Partially ordered ring – Ring with a compatible partial order
• Partially ordered space – Partially ordered topological space

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• Kopytov, V.M. (2001) [1994], "Partially ordered group", Encyclopedia of Mathematics, EMS Press
• Kopytov, V.M. (2001) [1994], "Lattice-ordered group", Encyclopedia of Mathematics, EMS Press
• This article incorporates material from partially ordered group on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.