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 submonoids that 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 $X$ .