partially ordered group

A partially ordered group is a group G that is a poset at the same time, such that if a,bG and ab, then

  1. 1.

    acbc, and

  2. 2.


for any cG. The two conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the one condition cadcbd for all c,dG. A partially ordered group is also called a po-group for short.


  • One of the immediate properties of a po-group is this: if ab, then b-1a-1. To see this, left multiply by the first inequality by a-1 on both sides to obtain ea-1b. Then right multiply the resulting inequality on both sides by b-1 to obtain the desired inequality: b-1a-1.

  • If can be seen that for every aG, the automorphismsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath La,Ra:GG also preserve order, and hence are order automorphisms as well. For instance, if bc, then La(b)=abac=La(c).

  • A element a in a po-group G is said to be positive if ea, where e is the identity elementMathworldPlanetmath of G. The set of positive elements in G is called the positive conePlanetmathPlanetmath of G.

  • (special po-groups)

    1. (a)

      A po-group whose underlying poset is a directed setMathworldPlanetmath is called a directed group.

      • *

        If G is a directed group, then G is also a filtered set: if a,bG, then there is a cG such that ac and bc, so that ac-1ba and ac-1bb as well.

      • *

        Also, if G is directed, then G=G+: for any xG, let a be the upper bound of {x,e} and let b=ax-1. Then eb and x=a-1bG+.

    2. (b)

      A po-group whose underlying poset is a latticeMathworldPlanetmath is called a lattice ordered group, or an l-group.

    3. (c)

      If the partial orderMathworldPlanetmath on a po-group G is a linear order, then G is called a totally ordered group, or simply an ordered group.

    4. (d)

      A po-group is said to be ArchimedeanPlanetmathPlanetmathPlanetmath if anb for all n, then a=e. Equivalently, if ae, then for any bG, there is some n such that b<an. This is a generalizationPlanetmathPlanetmath of the Archimedean property on the reals: if r, then there is some n such that r<n. To see this, pick b=r, and a=1.

    5. (e)

      A po-group is said to be integrally closedMathworldPlanetmath if anb for all n1, then ae. An integrally closed group is Archimedean: if anb for all n, then ae and eb. Since we also have (a-1)-nb for all n<0, this implies a-1e, or ea. Hence a=e. In fact, an directed integrally closed group is an AbelianMathworldPlanetmath po-group.

  • Since the definition above does not involve any specific group axioms, one can more generally introduce partial ordering on a semigroupPlanetmathPlanetmath in the same fashion. The result is called a partially ordered semigroup, or a po-semigroup for short. A lattice ordered semigroup is defined similarly.

Title partially ordered group
Canonical name PartiallyOrderedGroup
Date of creation 2013-03-22 16:42:25
Last modified on 2013-03-22 16:42:25
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Definition
Classification msc 06F05
Classification msc 06F20
Classification msc 06F15
Classification msc 20F60
Synonym po-group
Synonym l-group
Synonym Archimedean po-group
Synonym integrally closed po-group
Synonym po-semigroup
Synonym lattice-ordered group
Synonym l-semigroup
Related topic OrderedGroup
Defines directed group
Defines positive element
Defines positive cone
Defines lattice ordered group
Defines Archimedean partially ordered group
Defines integrally closed group
Defines integrally closed partially ordered group
Defines partially ordered semigroup
Defines lattice ordered semigroup
Defines Archimedean