A homomorphism of monoids \(\varphi:P\to Q\) is said to be local if \(\varphi^{-1}(Q^\times) = P^\times\) or equivalently if \(\varphi^{-1}(Q\setminus Q^\times) = P\setminus P^\times\). For sharp monoids and in particular toric monoids this simply means that \(\varphi^{-1}(0) = 0\).
All monoid morphisms resulting from morphisms of log schemes are local.