I'm trying to understand whether the multiplicative monoid $ \left\{x^n \right\} _{n\geq 0}$ freely generated by $x$ or $1,x$. So for monoids, are "zeroth powers" included in generating sets or not?
2026-03-27 11:48:59.1774612139
Convention for generators of monoids - is $ \left\{x^n \right\} _{n\geq 0}$ freely generated by $x$?
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A monoid is supposed to have an identity element, which is with free generation considered as the empty string. Hence your example translates to the free monoid on the letter $x$.