"I still don't understand how one can express a concept such as monoid or group in, say, Haskell." Roughly the same way one can express the concept in mathemat

"I still don't understand how one can express a concept such as monoid or group in, say, Haskell." Roughly the same way one can express the concept in mathematics. You say "this set forms a monoid under that operation, with such-and-such identity element", and optionally include some argument (or, ideally, proof) that this is the case. In (traditional/most) math, that is only checked by humans. In Haskell, it is checked by humans and hopefully tests (laws often make great QuickCheck properties!).