If we consider the exponential notation used in abstract algebra, we get a nice mnemonic for semigroups, monoids, and groups:

- Semigroup: Only
*positive*powers are defined - Monoid: The identity element makes \(a^0 = e\) defined, so
*non-negative*powers are defined - Group: inverses define negative powers, so all
*integer*powers are defined

By remembering the completion of the natural numbers into the integers, we can remember what exactly defines semigroups, monoids, and groups (I know I mix them up all the time).