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).