Semigroups

Okay kiddo, imagine you have a box of toys. You can combine any two toys inside the box to make a cool new toy. That's what we call a semigroup.

A semigroup is a set of things that you can combine together in a specific way. Just like with toys, there are some rules for combining things in a semigroup. The combination of two things in a semigroup is called an "operation".

One important rule is that the operation is always "associative." That's just a big word that means it doesn't matter in what order you combine things - you'll always end up with the same result.

For example, imagine you have some numbers and you want to combine them. If the two numbers are a and b, then combining them means doing some kind of operation like addition or multiplication. Associativity means that (a + b) + c is the same as a + (b + c), and (a x b) x c is the same as a x (b x c).

Another important rule of a semigroup is that there's no "identity" element. That means there's no one thing that you can combine with any other thing to get back the same thing.

So, by combining things in a semigroup, you can make all kinds of new things! Just like if you combine different toys in your toy box, you can make new, fun playthings. This can be helpful in all kinds of situations, like in math or computer science, where semigroups are used to make all kinds of algorithms and models.