If every group is a normal subgroup of itself, the normal subgroup in what does it become if I now have a semigroup?

Question

I know this

every group is a normal subgroup of itself
every group has identity element
every normal subgroup has identity element

The kernel of a group homomorphism $f:G\rightarrow G^{'}$ is the set of all elements of $G$ which are mapped to the identity element of $G^{'}$. The kernel is a normal subgroup of $G$, and always contains the identity element of $G$. It is reduced to the identity element iff f is injective.

Identity element is called also unit element or 1

What kind of structure do I have if I don’t have the identity element (unit element or 1) ?

I know that a group without identity element is a semigroup. My problem is to classify normal subgroup when I 'delete' identity element.

The normal subgroup in what does it become if I now have a semigroup?

Answer 1

A semigroup is just a group with no requirement of having an identity, or inverses. Simply "deleting" the identity in a group will not in general produce a semigroup. For example, the set $\mathbb{Z}\setminus\{0\}$ is not closed under addition.