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.

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