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Computationally intractable orbit of a monoid action on a finite set

Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$. A characterization of $M_n$ is an algorithm that takes an integer $...
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