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.


* 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".


*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.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Cyclically ordered group — In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order. Cyclically ordered groups were first studied in depth by Ladislav Rieger… …   Wikipedia

  • Partially-ordered group — In abstract algebra, a partially ordered group is a group (G,+) equipped with a partial order ≤ that 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. An element x… …   Wikipedia

  • Linearly ordered group — In abstract algebra a linearly ordered or totally ordered group is an ordered group G such that the order relation le; is total. This means that the following statements hold for all a , b , c isin; G :* if a le; b and b le; a then a = b… …   Wikipedia

  • Ordered semigroup — In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S. If S is a group and it is ordered… …   Wikipedia

  • Ordered pair — In mathematics, an ordered pair (a, b) is a pair of mathematical objects. In the ordered pair (a, b), the object a is called the first entry, and the object b the second entry of the pair. Alternatively, the objects are called the first and… …   Wikipedia

  • Group tournament ranking system — In a group tournament, unlike a knockout tournament, there is no decisive final match. Instead, all the competitors are ranked by examining the results of all the matches played in the tournament. Points are awarded for each fixture, with… …   Wikipedia

  • ordered — É”rdÉ™(r)d / ɔː adj. well arranged, neat; commanded, required or·der || É”rdÉ™(r) / ɔːd n. arrangement; instruction; command; request for something; religious group v. command; request something; arrange; manage …   English contemporary dictionary

  • Cyclic group — Group theory Group theory …   Wikipedia

  • Archimedean group — In abstract algebra, a branch of mathematics, an Archimedean group is an algebraic structure consisting of a set together with a binary operation and binary relation satisfying certain axioms detailed below. We can also say that an Archimedean… …   Wikipedia

  • Partially ordered set — The Hasse diagram of the set of all subsets of a three element set {x, y, z}, ordered by inclusion. In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitive concept of an ordering,… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.