Free monoid

Summary

The free monoid on a set $A$ is the monoid $A^*$ of all finite words over the alphabet $A$. Its operation is concatenation, and its identity is the empty word. It is free in the sense that any map from $A$ to a monoid extends in exactly one way to a monoid homomorphism from $A^*$. Free monoids are the setting of formal languages and automata, and they are the monoid analogue of free groups.