You can think of semigroups, monoids, and groups as the minimal structures needed to define and extend exponents.

All 3 structures are:

- a set
- an
*associative*binary operation on that set

Monoids and groups add an identity element.

Groups add inverses.

Now let’s see how exponents are a useful way to remember this.

## Associativity

This property is necessary for exponents to even make sense.

If , then is ambiguous. A good example is the binary operation of exponentiation with .

## Semigroups

Semigroups are sufficient to define strictly positive exponents, since is always well-defined.

## Monoids

Monoids let you define , where 1 is the identity element. So now you’ve added 0.

## Groups

If you define , then you get negative powers from the inverses.

## Takeaway

You can use the extension of the natural numbers to the integers by adding 0 and negative numbers to remember how semigroups, monoids, and groups are related.