Automorphism groups of mono-unary algebras and CH (Q2288415)

The author deals with the problem of finding all possible (infinite, since it is known that any finite value is possible) values of cardinal-valued functions \(\operatorname{card}\operatorname{Aut}(A, f )\) defined on the class of all mono-unary algebras \((A, f )\). The main result is as follows. For any infinite set \(A\), the following three statements are equivalent: \begin{itemize} \item[(1)] \(\vartheta\) is the cardinality of the automorphisms group of some partial mono-unary algebra whose universe is \(A\), \item[(2)] \(\vartheta\) is the cardinality of the automorphisms group of some mono-unary algebra whose universe is \(A\), \item[(3)] \(1 \leq \vartheta \leq \aleph_0\) or \(\vartheta=2\alpha\) for some cardinal number \(\alpha\leq|A|.\) \end{itemize} This result yields that the continuum hypothesis is equivalent to the condition that there exists a mono-unary algebra with exactly \(\aleph_1\) automorphisms. Next, three open problems are proposed.