Let $(S,*)$ be a semigroup. $(T,*)$ is a subsemigroup of $S$ if it is a semigroup and $T\subseteq S$.
Let $(S,*)$ be a semigroup. $T\subseteq S$ is a subsemigroup as soon as $T$ is closed under $*$, i.e. for all $a$, $b\in T$,
\[ab \in T.\]
Let $(S,*)$ be a semigroup. $(T,*)$ is a subsemigroup of $S$ if it is a semigroup and $T\subseteq S$.
Let $(S,*)$ be a semigroup. $T\subseteq S$ is a subsemigroup as soon as $T$ is closed under $*$, i.e. for all $a$, $b\in T$,
\[ab \in T.\]