Partially ordered group explained

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.

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.

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,...,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)

Properties

Archimedean

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

Property: A partially ordered group

is called Archimedean when for any

a,b\inG

, if

e\lea\leb

and

aⁿ\leb

for all

n\ge1

then

a=e

. Equivalently, when

a ≠ e

, then for any

b\inG

, there is some

n\inZ

such that

b<aⁿ

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.

References

M. Anderson and T. Feil, Lattice Ordered Groups: an Introduction, D. Reidel, 1988.
Birkhoff . Garrett . 1942. Lattice-Ordered Groups . The Annals of Mathematics . 43 . 2 . 313 . 10.2307/1968871 . 0003-486X.
M. R. Darnel, The Theory of Lattice-Ordered Groups, Lecture Notes in Pure and Applied Mathematics 187, Marcel Dekker, 1995.
L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, 1963.
Book: 10.1017/CBO9780511721243. Ordered Permutation Groups. 1982. Glass. A. M. W.. 9780521241908.
Book: 981449609X. [{{Google books|5oTVCgAAQBAJ|Partially Ordered Groups|page=191|plainurl=yes}} Partially Ordered Groups]. Glass. A. M. W.. 1999.
V. M. Kopytov and A. I. Kokorin (trans. by D. Louvish), Fully Ordered Groups, Halsted Press (John Wiley & Sons), 1974.
V. M. Kopytov and N. Ya. Medvedev, Right-ordered groups, Siberian School of Algebra and Logic, Consultants Bureau, 1996.
Book: 10.1007/978-94-015-8304-6. The Theory of Lattice-Ordered Groups. 1994. Kopytov. V. M.. Medvedev. N. Ya.. 978-90-481-4474-7.
R. B. Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.
Book: 10.1007/b139095. Lattices and Ordered Algebraic Structures. Universitext. 2005. 1-85233-905-5., chap. 9.
10.1016/0021-8693(76)90242-8. On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. 1976. Elliott. George A.. Journal of Algebra. 38. 29–44.