I haven't previously thought about this, but I think words over a commutative monoid are equivalent to a vector of non-negative integers, at which point you have vector addition systems, and I believe those are decidable, though still computationally incredibly hard: https://www.quantamagazine.org/an-easy-sounding-problem-yiel....

Thanks, that's an interesting tidbit!

(The whole thing made me think about applications to SQL query optimizers, although I'm not sure if it's practically useful for anything.)