partially ordered group

{{redirect|Ordered group|groups with a total or linear order|Linearly 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: {{nobreak|a ≤ b}} if and only if {{nobreak|-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 $e \le a \le b$ and $a^n \le b$ for all $n \ge 1$ 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.{{harvtxt|Glass|1999}}

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.{{harvtxt|Birkhoff|1942}}

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

References

M. Anderson and T. Feil, Lattice Ordered Groups: an Introduction, D. Reidel, 1988.
{{Cite journal |last=Birkhoff |first=Garrett |date=1942|title=Lattice-Ordered Groups |url=http://dx.doi.org/10.2307/1968871 |journal=The Annals of Mathematics |volume=43 |issue=2 |page=313 |doi=10.2307/1968871 |jstor=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.
{{cite book |doi=10.1017/CBO9780511721243|title=Ordered Permutation Groups|year=1982|last1=Glass|first1=A. M. W.|isbn=9780521241908}}
{{cite book |isbn=981449609X|title=Partially Ordered Groups|last1=Glass|first1=A. M. W.|year=1999|url={{Google books|5oTVCgAAQBAJ|Partially Ordered Groups|page=191|plainurl=yes}}}}
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.
{{cite book |doi=10.1007/978-94-015-8304-6|title=The Theory of Lattice-Ordered Groups|year=1994|last1=Kopytov|first1=V. M.|last2=Medvedev|first2=N. Ya.|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.
{{cite book |doi=10.1007/b139095|title=Lattices and Ordered Algebraic Structures|series=Universitext|year=2005|isbn=1-85233-905-5}}, chap. 9.
{{cite journal |doi=10.1016/0021-8693(76)90242-8|title=On the classification of inductive limits of sequences of semisimple finite-dimensional algebras|year=1976|last1=Elliott|first1=George A.|journal=Journal of Algebra|volume=38|pages=29–44|doi-access=}}

External links

{{Eom| title = Partially ordered group | author-last1 =Kopytov| author-first1 = V.M.| oldid = 48137}}
{{Eom| title = Lattice-ordered group | author-last1 =Kopytov| author-first1 = V.M.| oldid = 47589}}
{{PlanetMath attribution

|urlname=PartiallyOrderedGroup |title=partially ordered group

}}

Category:Ordered algebraic structures

Category:Ordered groups

Category:Order theory