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.