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 \((xx)x \neq x(xx)\), then \(x^3\) is ambiguous. A good example is the binary operation of exponentiation with \(x = 3\). \((3^3)^3 = 19,683 \neq 7,625,597,484,987 = 3^(3^3)\)

## Semigroups

Semigroups are sufficient to define strictly positive exponents, since \(x \cdot x\) is always well-defined.

## Monoids

Monoids let you define \(x^0 = 1\), where 1 is the identity element. So now you’ve added 0.

## Groups

If you define \(x^{-1} := inv(x)\), 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.