A log scheme consists of a scheme $X$, a sheaf of monoids $M_X$ on $X$, and a map $\alpha:M_X\to\mathcal O_X$ with the property that $\alpha^{-1}(\mathcal O_X^\times)\to\mathcal O_X^\times$ is an isomorphism.
Is there a standard term for the condition that the map $\alpha$ be injective?
I have looked in a number of standard sources and have not found any mention of this condition.
