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 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.
Examples 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,...,an) ≤ (b1,...,bn) if and only if ai ≤ bi (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)
Properties
Archimedean The Archimedean property of the real numbers can be generalized to partially ordered groups.
Property: A partially ordered group G {\displaystyle G} is called Archimedean when for any a , b ∈ G {\displaystyle a,b\in G} , if e ≤ a ≤ b {\displaystyle e\leq a\leq b} and a n ≤ b {\displaystyle a^{n}\leq b} for all n ≥ 1 {\displaystyle n\geq 1} then a = e {\displaystyle a=e} . Equivalently, when a ≠ e {\displaystyle a\neq e} , then for any b ∈ G {\displaystyle b\in G} , there is some n ∈ Z {\displaystyle n\in \mathbb {Z} } such that b < a n {\displaystyle b<a^{n}} .
Integrally closed 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. 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.
See also Cyclically ordered group – Group with a cyclic order respected by the group operation Linearly ordered group – Group with translationally invariant total order Ordered field – Algebraic object with an ordered structure Ordered ring 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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