These algebraic structures have sacrificed generality in favor of being easily used with the standard Haskell Prelude. The fact that monoids are not guaranteed to be semigroups makes this difficult.
Semigroups that also have an inverse. See https://en.wikipedia.org/wiki/Regular_semigroup
Regular semigroups that also have an identity; alternatively, monoids where every element has a unique inverse. See https://en.wikipedia.org/wiki/Group_(mathematics)