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.

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

G

is called Archimedean when for any

a,b\in G

, if

e\leq a\leq b

and

a^{n}\leq b

for all

n\geq 1

then

a=e

. Equivalently, when

a\neq e

, then for any

b\in G

, there is some

n\in \mathbb {Z}

such that

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.^[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]

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. doi:10.2307/1968871. ISSN 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.
Glass, A. M. W. (1982). Ordered Permutation Groups. doi:10.1017/CBO9780511721243. ISBN 9780521241908.
Glass, A. M. W. (1999). Partially Ordered Groups. ISBN 981449609X.
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.
Kopytov, V. M.; Medvedev, N. Ya. (1994). The Theory of Lattice-Ordered Groups. doi:10.1007/978-94-015-8304-6. ISBN 978-90-481-4474-7.
R. B. Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.
Lattices and Ordered Algebraic Structures. Universitext. 2005. doi:10.1007/b139095. ISBN 1-85233-905-5., chap. 9.
Elliott, George A. (1976). "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras". Journal of Algebra. 38: 29–44. doi:10.1016/0021-8693(76)90242-8.

External links

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.

[Glass-1] Glass (1999)

[2] Birkhoff (1942)

[1]

[2]