## Exhibit a subsemigroup of a cyclic semigroup which is not cyclic

Exhibit a subsemigroup of a cyclic semigroup which is not cyclic.

Consider the semigroup $\mathbb{N}$ of natural numbers under addition. Certainly $\mathbb{N}$ is cyclic with generator 1. Define $T \subseteq \mathbb{N}$ by $T = \{2a+3b\ |\ a,b \in \mathbb{N}\}$.

Certainly $T$ is nonempty, since $0 = 2\cdot 0 + 3 \cdot 0 \in T$. Moreover, if $2a_1 + 3b_1, 2a_2 + 3b_2 \in T$, then $(2a_1+3b_1) + (2a_2 + 3b_2) = 2(a_1+a_2) + 3(b_1+b_2) \in T$. So $T \subseteq \mathbb{N}$ is a subsemigroup of a cyclic semigroup.

Now suppose $T$ is cyclic, with generator $2a_0 + 3b_0$. Now $2 = 2 \cdot 1 + 3 \cdot 0 \in T$, so $2 = k(2a_0 + 3b_0)$ for some $k \in \mathbb{N}$. Certainly $k \neq 0$. If $b_0 \neq 0$, then we have $2 = k(2a_0+3b_0) \geq 3kb_0$ $\geq 3 > 2$, a contradiction. So $b_0 = 0$, and we have $2 = 2ka_0$. So $1 = ka_0$, and thus $a_0 = 1$. That is, $T$ is generated by $2 \cdot 1 + 3 \cdot 0 = 2$. But we also have $3 = 2 \cdot 0 + 3 \cdot 1 \in T$, so that $3 = 2k$ for some $k \in \mathbb{N}$, a contradiction.

So $T$ is not cyclic.