[Note: Some diagrams below cannot be displayed correctly in the web version. Please use the PDF version for correct display.]
Let \(S\) be a monoid. A right \(S\)-act \(A_S\) is called weakly pullback flat if the functor \(A_S\otimes-\) preserves all pullbacks of the form \[ \begin{tikzcd} _SP \ar[r,"p_1"] \ar[d,"p_2" '] & _SS \ar[d,"f"]\\ _SS \ar[r,"g" '] & _SS \rlap{\, .} \end{tikzcd} \] For this property condition \((O)\) has also been used. It is recalled that: strongly flat \(\Rightarrow\) weakly pullback flat \(\Rightarrow\) condition \((P)\). The authors prove “All cyclic right \(S\)-acts satisfying condition \((P)\) are weakly pullback flat if and only if every right reversible submonoid of \(S\) is weakly left collapsible” and “All weakly pullback flat cyclic right \(S\)-acts are strong flat if and only if every right reversible and weakly left collapsible submonoid of \(S\) is left collapsible”.
A monoid \(S\) is said to be right reversible if for any \(s,t\in S\) there exist \(u,v\in S\) such that \(us=vt\) and a monoid \(P\) is said to be left collapsible if for any \(p,q\in P\) there exists \(r\in P\) such that \(rp=rq\). A submonoid \(P\subseteq S\) is called weakly left collapsible if \[ (\forall s,s'\in P)\;(\forall z\in S)\;(sz=s'z\Rightarrow(\exists u\in P)\;(us=us')). \]