Why is a positive cone defined the way it is? Let $K$ be a field. Consider $P \subseteq K$ satisfying the following properties:

  
*
  
*$x \in P$ and $y \in P$ imply $x+y \in P$ and $xy \in P$
  
*$x \in K$ imply $x^2 \in P$
  
*${-1} \notin P$
  

We call such $P$ a prepositive cone. We further call $P$ a positive cone if $K = P \cup {-P}$.
I have a few questions regarding the notion of a positive cone (I have not found references for this concept elsewhere other than Wikipedia). 


*

*Why the name positive "cone"? I understand why positive is there, but not the "cone" part.

*If we keep Property $1$, delete Property $2$ and Property $3$, and add Property $2'$, which says that $x \in P$, or $x = 0$, or $x \notin P$ (only one case holds for any $x \in K$), and define a positive cone this way, what do we lose? Why not define positive cones this way instead?


Partial answer:

A set $C$ is called a “cone” with vertex at the origin if for any $x\in C$ and any scalar $a\ge0$, $ax\in C$.

Look familiar? There is a related concept of a cone for linear algebra.

A cone $C$ is a convex cone if $ax + by$ belongs to $C$, for any positive scalars $a,b$ and any vectors $x, y$ in $C$.

That should also look familiar.
If you look at the Wikipedia page, you’ll see these actually look like cones.


Convex cone generated by the conic combination of three black vectors.

