Free Monoid

A free monoid is a monoid constructed from an alphabet $A$. The set is the set of strings $A^{\ast }$, and the operation $+$ is concatenation.

Then $(A^{\ast },{+}_A)$ is a free monoid.

A monoid $(X,\circ )$ is free iff there is an alphabet $A$ s.t. $(A^{\ast },{+}_A)\cong (X,\circ )$.