Ordered group

From Wikipedia, the free encyclopedia

In abstract algebra, an ordered group is a group (G,+) equipped with a partial order "≤" which 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. Note that sometimes the term ordered group is used for a linearly (or totally) ordered group, and what we describe here is called a partially ordered group.

An element x of G is called positive element if 0 ≤x. The set of elements 0 ≤x is often denoted with G⁺, and it is called the positive cone of G. So we have a ≤ b if and only if -a+b ∈ G⁺.

By the definition, we can reduce the partial order to a monadic property: a ≤ b if and only if 0 ≤ -a+b. .

For the general group G, the existence of a positive cone specifies an order on G. A group G is an ordered 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

An ordered group G with positive cone G⁺ is said to be unperforated if n · g ∈ G⁺ for some natural number 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, we speak of a linearly ordered group. If the order on the group is a lattice order, i.e. any two elements have a least upper bound, it is a lattice ordered group.

A Riesz group is a unperforated 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 ordered groups, a map from G to H is a morphism of ordered groups if it is both a group homomorphism and a monotonic function. The ordered groups, together with this notion of morphism, form a category.

Ordered groups are used in the definition of valuations of fields.

[edit] Examples

• An ordered vector space is an ordered group
• A Riesz space is a lattice ordered group
• A typical example of an 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 an ordered group and X is some set, then the set of all functions from X to G is again an ordered group: all operations are performed componentwise. Furthermore, every subgroup of G is an ordered group: it inherits the order from G.

[edit] References

• M. Anderson and T. Feil, Lattice Ordered Groups: an Introduction, D. Reidel, 1988.
• 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.
• A. M. W. Glass, Ordered Permutation Groups, London Math. Soc. Lecture Notes Series 55, Cambridge U. Press, 1981.
• 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.
• V. M. Kopytov and N. Ya. Medvedev, The Theory of Lattice-Ordered Groups, Mathematics and its Applications 307, Kluwer Academic Publishers, 1994.
• R. B. Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.