ordered group

Definition 1.  We say that the subsemigroup S of the group G (with the operationMathworldPlanetmath denoted multiplicatively) defines an G, if

  • a-1SaSaG,

  • G=S{1}S-1   where  S-1={s-1:sS}  and the members of the union are pairwise disjoint.

The order “<” of the group G is explicitly given by setting in G:


Then we speak of the ordered group(G,<),  or simply G.

Theorem 1.

The order “<” defined by the subsemigroup S of the group G has the following properties.

  1. 1.

    For all  a,bG, exactly one of the conditions   a<b,a=b,b<a   holds.

  2. 2.


  3. 3.


  4. 4.


  5. 5.


  6. 6.


Definition 2.  The set G is an ordered group equipped with zero 0, if the set G* of its elements distinct from its element 0 forms an ordered group  (G*,<)  and if

  • 0a=a0=0aG,

  • 0<aaG*.


  • 1 Emil Artin: Theory of Algebraic NumbersMathworldPlanetmath.  Lecture notes.  Mathematisches Institut, Göttingen (1959).
  • 2 Paul Jaffard: Les systèmes d’idéaux.  Dunod, Paris (1960).
Title ordered group
Canonical name OrderedGroup
Date of creation 2013-03-22 14:54:36
Last modified on 2013-03-22 14:54:36
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Definition
Classification msc 06A05
Classification msc 20F60
Related topic KrullValuation
Related topic PartiallyOrderedGroup
Related topic PraeclarumTheorema
Defines ordered group equipped with zero