Picking up some of Eric Zaslow's reformulation: Assume $P$ is *commutative* and *cancellative*, as well as finitely generated.  (Cancellative means $p_1+q=p_2+q$ implies $p_1=p_2$.)  The answer to your question is affirmative if and only if the "groupification" $P^{gp}$ of $P$ is torsion-free.

All this amounts to $P$ being embeddable as a sub-monoid of ${\Bbb Z}^n$ for some $n$.  Then take the subgroup of ${\Bbb Z}^n$ spanned by $P$.  This is isomorphic to some ${\Bbb Z}^m$; take the dual of the convex hull of $P$ in ${\Bbb R}^m$ and you've got your cone $\sigma$, just as Eric says.

Depending on what references you use, when $P^{gp}$ is torsion-free, $P$ is called either *integral* or *toric*.  (See, e.g., the toric variety notes on M. Mustata's <a href="http://www.math.lsa.umich.edu/~mmustata/toric_var.html">webpage</a> versus the log geometry notes on Danny Gillam's  <a href="http://www.math.brown.edu/~wgillam/">webpage</a>; both sources are worth looking at.)  It seems the latter terminology is more standard in the log geometry world, where "integral" sometimes just means "cancellative".