I'm trying to understand the relation between the points of view of log geometry (monoids) and toric geometry (fans).
Suppose that $k$ is a field and $P$ is a finitely generated monoid. Then $k[P]$ has a natural log structure and furthermore, any choice of generators $\mathbf N^r\to P$ induces a closed embedding $Spec(k[P])\subset\mathbf A^r$.
On the other hand, starting from a cone $\sigma$ satisfying some properties in a lattice $N\otimes\mathbf R$, where $N = \mathbf Z^r$, we obtain a monoid $P' = \sigma^\vee\cap M$, where $M = Hom(N,\mathbf Z)$ and $\sigma^\vee$ is the set of all $x\in M\otimes\mathbf R$ such that $x(\sigma) \geq 0$.
Question: if we start with $P$ (and a choice of generators as above), can one write a corresponding cone so as to recover $P$ by the construction in the previous paragraph?