free hull

Let $A$ be an arbitrary set,let $A^{\ast }$ be the free monoid on $A$,and let $X$ be a subset of $A^{\ast }.$It follows from the characterization of free submonoidsthat the intersection $M$ of all the free submonoids of $A^{\ast }$that contain $X$ is a free submonoid of $A^{\ast }$.The minimal generating set $H$ of $M$is called the free hull of $X$.