On inverse limits of monounary algebras. (Q2877073)

This paper deals with inverse limits of monounary algebras. Let \(\mathcal T\) be a class of monounary algebras \(A\) such that every connected component of \(A\) is a cycle; and if \(C\neq D\) are two components, then \(|C|\) is not a multiple of \(|D|\). It is proved that the singleton \(\{A\}\) is an inverse limit closed class if and only if \(A\in\mathcal T\) or \(A\) is isomorphic to the set of all integers or \(A\) is isomorphic to the set of all positive integers, both with the successor operation. Other types of inverse limit closed classes of monounary algebras are also described. Moreover, it is proved that \(\underleftarrow{\mathbf L}\underleftarrow{\mathbf L}\mathcal T\neq\underleftarrow{\mathbf L}\mathcal T\), where \(\underleftarrow{\mathbf L}\mathcal T\) denotes the class of all isomorphic copies of inverse limits of algebras from \(\mathcal T\).