Let $S$ be a finite semigroup and $T \subseteq S$ which satisfy the following property:
For $x, y \in T$, we have $x, y \in \langle z \rangle$ for some $z \in S$.
If $H \subseteq S$ satisfy the above property and $T \subseteq H$, then $H = T$.
Is $T$ forms a subsemigroup of $S$.
I have tried to find out the counter example of a semigroup $S$ and a subset $T$ of $S$ which satisfies these two properties but it does not form a subsemigroup of $S$. But I am unable to find this. After that, I have tried to prove $T$ forms a subsemigroup. It is sufficient to prove for any $a, b, c \in T$, there exist $d \in S$ such that $ab, c \in \langle d \rangle.$ From the given hhypothesis, there exist $x, y, z \in S$ such that $a,b \in \langle x \rangle$, $a, c \in \langle y \rangle$, $b,c \in \langle z \rangle$. I am stuck here. Thanks for your any kind of help.