Editing Linearly ordered group (section)

== Further definitions ==

{{more footnotes needed|section|date=July 2024}}

In this section <math>\le</math> is a left-invariant order on a group <math>G</math> with [[identity element]] <math>e</math>. All that is said applies to right-invariant orders with the obvious modifications. Note that <math>\le</math> being left-invariant is equivalent to the order <math>\le'</math> defined by <math>g \le' h</matH> if and only if <math>h^{-1} \le g^{-1}</math> being right-invariant. In particular a group being left-orderable is the same as it being right-orderable. 

In analogy with ordinary numbers we call an element <math>g \not= e</math> of an ordered group '''positive''' if <math>e \le g</math>. The set of positive elements in an ordered group is called the '''positive cone''', it is often denoted with <math>G_+</math>; the slightly different notation <math>G^+</math> is used for the positive cone together with the identity element.{{sfn|Deroin|Navas|Rivas|2014|loc=1.1.1}} 

The positive cone <math>G_+</math> characterises the order <math>\le</math>; indeed, by left-invariance we see that <math>g \le h</math> if and only if <math>g^{-1} h \in G_+</math>. In fact a left-ordered group can be defined as a group <math>G</math> together with a subset <math>P</math> satisfying the two conditions that:
#for <math>g, h \in P</math> we have also <math>gh \in P</math>;
#let <math>P^{-1} = \{g^{-1}, g \in P\}</math>, then <math>G</math> is the [[disjoint union]] of <math>P, P^{-1}</math> and <math>\{e\}</math>. 
The order <math>\le_P</math> associated with <math>P</math> is defined by <math>g \le_P h \Leftrightarrow g^{-1} h \in P</math>; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of <math>\le_P</math> is <math>P</math>.

The left-invariant order <math>\le</math> is bi-invariant if and only if it is conjugacy invariant, that is if <math>g \le h</math> then for any <math>x \in G</math> we have <math>xgx^{-1} \le xhx^{-1}</math> as well. This is equivalent to the positive cone being stable under [[inner automorphism]]s. 


If <math>a \in G</math>{{cn|date=July 2024}}, then the '''absolute value''' of <math>a</math>, denoted by <math>|a|</math>, is defined to be: <math display=block>|a|:=\begin{cases}a, & \text{if }a \ge 0,\\ -a, & \text{otherwise}.\end{cases}</math>
If in addition the group <math>G</math> is [[abelian group|abelian]], then for any <math>a, b \in G</math> a [[triangle inequality]] is satisfied: <math>|a+b| \le |a|+|b|</math>.