On some open problems of Mahmoudi and Renshaw. (Q2877804)

For a word \(w\) in an alphabet \(X\), \(C(w)\) denotes the set of elements of \(X\) appearing in \(w\). For a set of words \(R\), \(C(R)\) is defined as \(\bigcup_{w\in R}C(w)\). Let \(X\) be a set with more than two elements, \(S\) the free monoid generated by \(X\) and \(\rho\) a right congruence on \(S\). It is proved that the cyclic right \(S\)-act \(S/\rho\) has: 1) a (P)-cover if and only if \(|C([1]_\rho)|\in\{0,1\}\), and 2) a strongly flat cover if and only if \(|C([1]_\rho)|=0\). Examples are constructed showing that for any cardinal number \(\alpha\), there exists a cyclic \(S\)-act \(A\) such that the number of (P)-covers (respectively, strongly flat covers) of \(A\) is \(\alpha\). An example of a cyclic \(S\)-act is given that does not have a (P)-cover.