Ordered group

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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 ag ≤ bg and ga ≤ gb. 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.

By the definition we can reduce the partial order to a monadic property: a ≤ b if and only if 1 ≤ a^-1 b. The set of elements x ≥ 1 of G are often denoted with G⁺, and it is called the positive cone of G. So we have a ≤ b if and only if a^-1b ∈ G⁺. That G is an ordered group can be expressed only in terms of G⁺: A group is an ordered group if and only if there exists a subset H (which is G⁺) of G such that:

• 1 ∈ H
• if a ∈ H and b ∈ H then ab ∈ H
• if a ∈ H then x^-1ax ∈ H for each x of G
• if a ∈ H and a^-1 ∈ H then a=1

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, we speak of a lattice ordered group.

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

• A 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.