A monoid is a semigroup with an identity element, and as such inherits much of its behavior from semigroups.

For historical reasons, a Haskell Monoid is not based on a notional Haskell Semigroup. There are packages that add semigroups and other algebraic structures, and even alternative Preludes that provide a reasonably complete set of algebraic structures. Every so often you'll see bikeshedding in the Haskell community over whether the default Prelude should provide some or all of these. :)