Given a convex cone $P\subset X$ where $X$ is a $K$-vector space, $K=\mathbb{R}\text{ or }\mathbb{C}$ is a field.
Suppose that $P$ satisfies positive element stipulations.
(1) $X=P-P$.
(2) $P\cap-P=\{0\}$.
Then we can define a partial ordering on $X$ by defining $x\geq_P y\Leftrightarrow x-y\in P$ for $\forall x,y\in X$.
Now we construct a norm $\left\Vert \cdot\right\Vert$ on this space $X$, if the closure w.r.t. norm induced topology $P\subsetneqq\bar{P}\subset X$ also satisfies positive element stipulations, is there an example that $\left(X,\geq_{P}\right)$ and $\left(X,\geq_{\bar{P}}\right)$ are both vector lattices?
