A remark: from a programming perspective, monoids are not esoteric, and are indeed more important than any of the intro abstract algebra structures. This is because free monoids are ordered lists, the most fundamental data structure in all of programming.
You can have a long and productive career as a C programmer knowing nothing but how to create and iterate over arrays! Monoids are so important that every single modern language (from C++ and Java all the way to Haskell and Scala) devotes a HUGE amount of language and standard library surface area to render data structures as sequences.
They are so important that even a language like Haskell is willing to accept utterly nonsensical semantics (hello, Traversable) to better support sequences.
> Monoids are so important that every single modern language (from C++ and Java all the way to Haskell and Scala) devotes a HUGE amount of language and standard library surface area to render data structures as sequences.
Yes and no. Yes, they devote a huge amount of language and standard library surface area to sequences. No, nobody in C cares about that being a monoid, or is thinking about monoids when they use sequences, just like no NBA players think about general relativity when they're shooting a basketball. With C++... maybe a few think about it that way.
Well, I think unsigned integers are even more fundamental than ordered lists, and those behave as the ring of integers mod 2^bit_width .
So aren't rings even more important in programming than monoids?
My answer would be that neither really is.
You're confusing a few examples of important things happening to be X, with the concept of X being an important concept.
When people use ordered lists, they are not thinking of them as monoids, or using any interesting theorems from monoid theory.
If mathematicians had never come up with the definition of monoids, C and C++ would have been designed in exactly the same way as they are now.
I've mostly ignored Haskell for the past decade. What happened with Traversable?