Ordered group

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 "x"1, "x"2, "y"1, "y"2 are elements of "G" and "xi" ≤ "yj", then there exists "z" ∈ "G" such that "xi" ≤ "z" ≤ "yj".

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.

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 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, every subgroup 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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