# Ordered group

In

abstract algebra , an**ordered group**is a group "(G,+)" equipped with apartial 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 alinearly 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 "x"_{1}, "x"_{2}, "y"_{1}, "y"_{2}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 amonotonic function . The ordered groups, together with this notion of morphism, form a category.Ordered groups are used in the definition of valuations of fields.

**Examples*** An

ordered vector space is an ordered group

* ARiesz space is a lattice ordered group

* A typical example of an ordered group is**Z**^{"n"}, where the group operation is componentwise addition, and we write ("a"_{1},...,"a"_{"n"}) ≤ ("b"_{1},...,"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, everysubgroup of "G" is an ordered group: it inherits the order from "G".**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.

*T.S. Blyth, "Lattices and Ordered Algebraic Structures", Springer, 2005, ISBN 1-85233-905-5, chap. 9.

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