Generalizations of free monoids

Abstract: We generalize free monoids by defining $k$-monoids. These are nothing other than the one-vertex higher-rank graphs used in $C^{\ast}$-algebra theory with the cardinality requirement waived. The $1$-monoids are precisely the free monoids. We then take the next step and generalize $k$-monoids in such a way that self-similar group actions yield monoids of this type.