Let $S$ be a finite semigroup and $T$ be a subset of $S$ which satisfy the property for any $x, y \in T$, there exist $a \in S$ such that $x, y \in \langle a \rangle$. Is $\langle T \rangle$ a monogenic subsemigroup of $S$.
I have tried by induction, but I stuck. Any help would be appreciated. Thank you
The answer is no. Take $S = \{a, a^2, a^3 \}$ with $a^3 = a^4$ and $T = \{a^2, a^3\}$.
EDIT. As observed by user120386, this example has to be modified as follows: $S = \{a, a^2, a^3, a^4 \}$ with $a^4 = a^5$ and $T = \{a^2, a^3\}$. Then $\langle T \rangle = \{a^2, a^3, a^4\} \subseteq \langle a \rangle$, but $\langle a^2 \rangle = \{a^2, a^4 \}$ and $\langle a^3 \rangle = \{a^3, a^4 \}$, and thus $\langle T \rangle$ is not monogenic.