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.

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.