Poor choice of words on my part. Yes, monoids are indeed a very basic kind of algebraic structure but the tutorials for these structures are always in a language like haskell where the actual simplicity of the concepts gets lost behind the type system. Type systems in my mind are logical structures and even though there is a very close relationship between algebra and logic one doesn't always have to tie the algebraic structures to some kind of type system to make sense of them.
Nods. The nice part about the type system is that it can automatically reject (some) misuses of the structures. I've found that helpful while learning some of these things as it can let me know that I don't actually understand something that I thought I had. I agree that emphasizing the typed aspect can do an injustice to the other algebraic properties.
