Cones and partial ordering A positive cone always induces a partial ordering on a vector space. I wonder if the converse holds, so is every partial ordering in a vector space induced by a positive cone, or in other words, is there a bijection between positive cones and partial orderings?
For a partial ordering $\succ$ on a set $\mathcal{S}$, its algebraic dual, the set of $\succ$-monotones $\mathcal{F}$ is a conic partial ordering. Since $f_j \in \mathcal{F}$ admits the conic combination $ \sum_j \mu_j f_j \in \mathcal{F}$ with $\sum_j \mu_j = 1, \mu_j \geq 0$ so it is a cone. The conic partial ordering $\geq_c$ is defined by $f_j \geq_c f_k$ if and only if $f_j - f_k \in \mathcal{F}$. For some conic partial ordering $\mathcal{F}$ over $\mathcal{S}$, in turn, induces the partial ordering $\succ$. So it is true.
I think it is probability not true. Since you are asking for partial order, you can easily define one that does not preserve under scalar multiplication.

e.g.

define a ($\mathbb{R}, \geq$) such that 

(1). $x\geq y$ if $(x=1,y=0)$ or $(x=y)$

(2). $x=y$ if $(x=y)$. 

It should satisfied the three properties of partial order. But clearly it does not preserved under scalar multiplication. It should be easy to proof that no cone can deduct such partial order.