Editing Ordered field (section)

===Positive cone===
A '''{{visible anchor|prepositive cone}}''' or '''preordering''' of a field <math>F</math> is a [[subset]] <math>P \subseteq F</math> that has the following properties:<ref name=Lam289>Lam (2005) p. 289</ref>
* For <math>x</math> and <math>y</math> in <math>P,</math> both <math>x + y</math> and <math>x \cdot y</math> are in <math>P.</math>
* If <math>x \in F,</math> then <math>x^2 \in P.</math>  In particular, <math>0 = 0^2 \in P</math>  and  <math>1 = 1^2 \in P.</math>
* The element <math>- 1</math> is not in <math>P.</math>

A '''{{visible anchor|preordered field}}''' is a field equipped with a preordering <math>P.</math>  Its non-zero elements <math>P^*</math> form a [[subgroup]] of the multiplicative group of <math>F.</math>

If in addition, the set <math>F</math> is the union of <math>P</math> and <math>- P,</math> we call <math>P</math> a '''positive cone''' of <math>F.</math>  The non-zero elements of <math>P</math> are called the '''positive''' elements of <math>F.</math>

An ordered field is a field <math>F</math> together with a positive cone <math>P.</math>

The preorderings on <math>F</math> are precisely the intersections of families of positive cones on <math>F.</math>  The positive cones are the maximal preorderings.<ref name=Lam289/>