NOW WHAT IS ACTUALLY A SEMIGROUP?

As I read, a semigroup is nothing other than an associative groupoid. It is said that a semigroup does not have to have an identity. Now I am curious to know what is a groupoid and why there does not need to exist an identity element in a semigroup and what are they or how they look. As far as inverses is concern there need to be none. Thus the set of integers will not qualify here. for every element in the integers with the ecception of zero has an inverse and as we saw earlier, the set of the integers is a ring and futhermore an integral domain. So semigroups must be sets with no identity and no inverses. I am thinking of the set of natural numbers. Natural numbers have no inverses and we know that the associative propety will apply for addition. In the natural numbers there exist no identity for addition because the identity for addition is zero and zero only belongs to the whole nubmers, yet something else must a a semigroup with no inverses and no identity property. I will have to wait to see more on examples of semigroup to see their truly importance, but for now I am satisfy with how much I have undertood so far about groups, rings, semigroups, and on idempotents. My worry so far is knowing more about groups. As far as I can understand a semigroup with an identity is a groupoid, and the natural numbers under the operation of addition while it seems to be a semigroup is not a monoid.