Definition 1. Let $(S, *)$ be a semigroup. $(T, *)$ is a subsemigroup of $(S, *)$, written $S \leq T$, if $(S, *)$ itself is a semigroup and $S \subseteq T$.
Proposition 1 (Subsemigroup Test). Let $(S, *)$ be a semigroup. $T \subseteq S$ is a subsemigroup as soon as $T$ is closed under $*$, i.e. for all $a$, $b \in T$,
\[ab \in T.\]