We generalize free monoids by defining k-monoids. These are nothing other than the one-vertex higher-rank graphs used in C^-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.
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Lawson et al. (2024) studied this question.
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