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. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...

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 is 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⁺. ↔ ⇔ ≡ logical symbols representing iff. ...

The order of an ordered group G is defined by G⁺; a group is an ordered group if and only if there exists a subset H (which is G⁺) of G such that: ↔ ⇔ ≡ logical symbols representing iff. ...

• 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. In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, a linearly ordered group is both a group and a linearly ordered set, in which the group operation is in a certain sense compatible with the linear ordering. ...

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. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

Ordered groups are used in the definition of valuations of fields. Model Theory In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

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.

A point x in R2 and the set of all y such that x≤y (in red). ... In mathematics a Riesz space, lattice-ordered vector space or vector lattice is an ordered vector space where the order structure is a lattice. ... The integers are commonly denoted by the above symbol. ... ↔ ⇔ ≡ logical symbols representing iff. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

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.