I did. I left it out for two reasons:
- the comment asked for a "simpler" explanation and associativity isn't a specially simple concept if you are not familiar with it and does not really help understand the core idea of a monoid.
- in practice, monoid associativity isn't checked. Most type system aren't able to express it. Therefore, whenever you encounter a monoid you have no guarantee that it is indeed associative.
There may be no guarantee, but that doesn't make a structure without associativity any more of a monoid, nor is having an identity element without associativity the core of idea of a monoid. If anything, associativity is the more important part of the concept since without an identity element, you are still left with a semigroup.
