Generalizations of free monoids

We generalize free monoids by defining k-monoids. These are nothing other than the one-vertex higher-rank graphs used in \(C^{*}\) $C_{}^{\ast }$ -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.