Are the multiplicative inverses contained in the cone?

Question

Definition: A cone of a field $F$ is a subset $P$ of $F$ such that :

1. $x,y\in P\implies x+y\in P$
2. $x,y\in P\implies xy\in P$
3. $x\in F\implies x^2\in P$

I would like to know if the third condition is important in this definition, because for me it's just a consequence of the second one. Another doubt is if $x^{-1}\in P$ whenever $x\in P$.

Thanks a lot!

Answer 1

Point 3 is not just a consequence of point 2. For example, $\Bbb Z$ satisfies 1 and 2 in $\Bbb Q$, but it does not satisfy 3.

Point 3 tells you that for any $x\in P$, $x^{-2}\in P$. Multipying $xx^{-2}$, you get $x^{-1}\in P$.

Answer 2

The example you should go to is $\mathbb R^+=\{x\in \mathbb R\mid x\geq 0\}$. This is a cone of $\mathbb R$ since $x+y\geq 0$ and $xy\geq 0$ if $x\geq 0$ and $y\geq 0$, and if $x\in \mathbb R$ then $x^2\in \mathbb R^+$. This should clarify that (2) and (3) are distinct conditions: we require that (for example) $(-1)^2\in \mathbb R^+$. In particular, a set like $\{x\in \mathbb R\mid x\geq 2\}$ is not a cone even though it satisfies (1) and (2).