Exhibit a semigroup of transformations having no idempotents

Give an example of a semigroup of transformations having no idempotents.


We saw in this previous exercise that every finite semigroup contains idempotent elements. So any example we find will have to be transfinite.

Given k \in \mathbb{N}^+, define \varphi_k : \mathbb{N}^+ \rightarrow \mathbb{N}^+ by \varphi_k(a) = a+k. Let S \subseteq \mathsf{T}(\mathbb{N}^+) be the subset S = \{\varphi_k\ |\ k \in \mathbb{N}^+\}. We claim that S is a subsemigroup of the full semigroup of transformations on \mathbb{N}^+; indeed, for all k,\ell \in \mathbb{N}^+ and a \in \mathbb{N}^+, we have (\varphi_k \circ \varphi_\ell)(a) = \varphi_k(\varphi_\ell(a)) = \varphi_k(a + \ell) = a+\ell+k = \varphi(\ell+k)(a), so that S is closed under composition. So S is a semigroup of transformations.

If \varphi_k is idempotent, then a+k = \varphi_k(a) = (\varphi_k \circ \varphi_k)(a) = a+2k for all a \in \mathbb{N}^+, and so k = 2k. This equation has no solutions in \mathbb{N}^+, so that S does not contain an idempotent.

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