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.
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 are sufficient to define strictly positive exponents, since is always well-defined.
Monoids let you define , where 1 is the identity element. So now you’ve added 0.
If you define , then you get negative powers from the inverses.
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.