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.