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I recently found this theorem when I am studying Semigroup presentations.


Theorem: If S is a semigroup generated by denumerably many elements, then S can be embedded into a semigroup generated by two elements.


As a hint for constructing the proof a theorem was given:

Theorem: Let $\alpha_{1},\alpha_{2}...:\mathbb N_{+}\to \mathbb N_{+}$ be any transformations. There exist two transformation $\beta_{1},\beta_{2}:\mathbb N_{+}\to \mathbb N_{+}$ such that each $\alpha_{i}$ is a composition of these two transformations.

By using a suitable isomorphisms Evans' theorem follows from the above result. But, to my suprise constructing the proof is given me headache. Please can someone assist?

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    Just to note that the theorem you mention is an old result of Sierpi\'nski from 1935:2013-11-30

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