Homoflatness on ideal extensions. (Q1780026)

The article considers acts over monoids that are of type \(S=G\dot\cup I\) where \(G\) is a group and \(I\) an ideal of \(S\); the monoid \(I\cup 1\) is denoted by \(I^1\). A right \(S\)-act \(A_S\) is called `weakly homoflat' (`principally weakly homoflat') if tensoring by \(A_S\) of an arbitrary pullback diagram \(P(I,I,f,f,S)\) (\(P(Ss,Ss,f,f,S)\), respectively), \(I\) a left ideal of \(S\) and \(f\colon I\to S\) a homomorphism, the corresponding canonical mapping \(\varphi\colon A_S\otimes_SP\to P'\) is surjective (some authors are using the notations (WP) and (PWP) for these properties, respectively). The main results show that if a right \(S\)-act \(A_S\) is weakly homoflat (principally weakly homoflat) as a right \(I^1\)-act, then it is weakly homoflat (principally weakly homoflat, respectively) as a right \(S\)-act. An example is given showing that a similar result is not true for the following properties: (weak) pullback flatness, equalizer flatness, (principally) weak kernel flatness, translation kernel flatness, condition (\(E'\)).