I've been reading Spivak and Fong's "An invitation to applied category theory" and I read that a monoid is a function $*$, called monoid multiplication for which:
- There is a specific element $I$ in the function domain, called monoidal unit, for which unitality holds
- The operation is associative (i.e. $*(*(a,b),c) = *(a, *(b, c))$ for all $a,b,c$ in the domain
I also read that a monoid can support commutativity. Then, there are symmetric monoidal preorders which, to my understanding, are commutative (aka symmetric) monoids in which the function domain is a preorder, and an extra property (monotonicity) holds.
My doubt is, don't the terms symmetric monoidal and commutative monoid mean the same thing, when they're clearly not? To be precise, the former doesn't mention in any way the monotonicity, almost as if it were implied that having a commutative monoid in which the domain is a preorder grants monotonicity, but this clearly isn't the case. Is there an explanation for this unfortunate naming?