I'm too scared to leave the comfy world of commutative monoids.

Is the word problem easier if the monoids are commutative? (Or even trivial? I haven't thought deeply about it.)

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....

	▲	Sesse__ 3 hours ago \| parent [-]
		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.)