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Is it possible that a convex cone and its closure both induce vector lattices?

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=...
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