Ones in subsemigroups

Question

I found the following definition in a PDF.

Definition 2.1. Let S be a semigroup and ∅ 6= T ⊆ S. Then T is a subsemigroup of S if a, b ∈ T ⇒ ab ∈ T. If S is a monoid then T is a submonoid of S if T is a subsemigroup and 1 ∈ T. Note T is then itself a semigroup/monoid.

I want to know whether there exist subsemigroups where the identity of the subsemigroup is different from the identity of the larger semigroup (as can happen in matrix rings, where we have an identity matrix with some zeroes along the main diagonal in a subring).

If so, then, is this definition wrong?

Thanks.

Answer 1

Let $e$ be an idempotent of a monoid $M$. Then $\{e\}$ is a subsemigroup of $M$. It is also a monoid with identity $e$, but it is not a submonoid of $M$ if $e \not= 1$.