Semigroup with two elements

Hello there! Have you ever played with two toys and pretended they were best friends? Well, sometimes we use something called a "semigroup" to talk about two things that can play together too!

A semigroup with two elements is like having two special toys we call "a" and "b". We can combine them in a certain way, but it doesn't matter which one we put first.

For example, we can put "a" and "b" together like this: "ab". And we could also do it the other way: "ba". But no matter which way we do it, we end up with the same thing. This is called the "associative property" of a semigroup.

So basically, a semigroup with two elements is just like having two toys that play together, and we can combine them in different ways but they always end up the same. Cool, huh?