Let \(\mathcal N\) be the semigroup \(\mathbb N^{\mathbb N}\) of all mappings on the set \(\mathbb N\) of all natural numbers and let \(\mathcal{P=P(N)}\) be the set of subsets of \(\mathcal N\). A preorder \(\preccurlyeq\) on \(\mathcal P\) is defined as follows: \(U\preccurlyeq V\) if there exists a countable subset \(C\) of \(\mathcal N\) such that \(U\) is contained in the semigroup \(\langle V,C\rangle\) generated by \(V\) and \(C\). \(U\approx V\) means that \(U\preccurlyeq V\) and \(V\preccurlyeq U\).
It is proved that the Continuum Hypothesis holds if and only if there exists a subsemigroup \(S\) of \(\mathcal N\) such that \(S\approx\mathcal N\) and for all subsemigroups \(T\) of \(S\) either \(T\approx\mathcal N\) or \(T\approx\{1_{\mathbb N}\}\).
Furthermore, the authors use the semigroups \(\mathfrak F_n=\{f\in\mathcal N:|f(\mathbb N)|\leq n\}\), \(n\geq 2\), and \(\mathfrak F=\bigcup_{n\in\mathbb N}\mathfrak F_n\) and their subsemigroups for describing the properties of the preorder \(\preccurlyeq\). For example, if \(S\) is a closed subsemigroup of \(\mathcal N\) of cardinality \(2^{\aleph_0}\) then there exists a closed subsemigroup \(T\) of \(\mathfrak F_2\) such that \(|T|=2^{\aleph_0}\) and \(T\preccurlyeq S\).