Covers of acts over monoids and pure epimorphisms. (Q2921037)

Let \(S\) be a monoid. An epimorphism \(\psi\colon X\to Y\) of right \(S\)-acts is called \textit{pure} if for every finitely presented \(S\)-act \(M\) and every \(S\)-map \(f\colon M\to Y\) there exists \(g\colon M\to X\) such that \(f=\psi g\). It is proved that an epimorphism of \(S\)-acts is pure if and only if it is a directed colimit of split epimorphisms.NEWLINENEWLINE Let \(\mathcal X\) be a class of \(S\)-acts closed under isomorphisms. The authors define an \(\mathcal X\)-\textit{precover} of an \(S\)-act \(A\) as an \(S\)-map \(g\colon P\to A\) for some \(P\in\mathcal X\) such that, for every \(S\)-map \(g'\colon P'\to A\), for \(P'\in\mathcal X\), there exists an \(S\)-map \(f\colon P'\to P\) with \(g'=gf\). If, in addition, the precover satisfies the condition that each \(S\)-map \(f\colon P\to P\) with \(gf=g\) is an isomorphism, then it is called an \(\mathcal X\)-\textit{cover}. If \(\mathcal X\) is also closed under directed colimits and if \(A\) is a right \(S\)-act that has an \(\mathcal X\)-precover, then \(A\) has an \(\mathcal X\)-cover. Necessary and sufficient conditions are found for \(\mathcal X\) where every \(S\)-act has an \(\mathcal X\)-precover (under the assumption that a coproduct \(\dot\cup_IX_i\in\mathcal X\Leftrightarrow X_i\in\mathcal X\) for each \(i\in I\)).NEWLINENEWLINE The results are applied in the cases where \(\mathcal X\) coincides with the class of strongly flat \(S\)-acts or with the class of acts satisfying condition (P). For example, in both cases, all right \(S\)-acts over the following monoids have an \(\mathcal X\)-cover: 1) finite monoids, 2) rectangular bands with a 1 adjoined, 3) right groups with a 1 adjoined, 4) right simple semigroups with a 1 adjoined, 5) \((\mathbb N,\max)\).