A monoid in which Levi's lemma holds is said to have the equidivisibility property. The free monoid of strings and string concatenation has this property (by Levi's lemma for strings), but by itself equidivisibility is not enough to guarantee that a monoid is free. However an equidivisibile monoid "M" is free if additionally there exists a homomorphism "f" from "M" to the monoid of natural numbers (free monoid on one generator) with the property that the preimage of 0 contains only the identity element of "M", i.e. formula_1. (Note that "f" simply being a homomorphism does not guarantee this latter property, as there could be multiple elements of "M" mapped to 0.) A monoid for which such a homomorphism exists is also called "graded" (and the "f" is called a gradation).