A characterization of piecewise \(\mathscr{F} \)-syndetic sets (Q6623520)

The author uses notions such as \((\mathscr{F},\mathscr{G})\)-syndeticity and piecewise \(\mathscr{F}\)-syndeticity, which were defined by \textit{O. Shuungula} et al. in their paper [Semigroup Forum 79, No. 3, 531--539 (2009; Zbl 1184.22001)]. These notions are generalizations of the well-known notions of syndeticity and piecewise syndeticity. In the paper, the author gives a characterization for an \(\mathscr{F}\)-piecewise syndetic set [Theorem 4.5]. He then answers in Theorem 4.10 affirmatively (using the mentioned characterization) to the question [Question 4.6] posed in the paper [\textit{C. Christopherson} and \textit{J. H. Johnson jun.}, Semigroup Forum 104, No. 1, 28--44 (2022; Zbl 1507.22008)], which reads as follows: ``Let \(\mathscr{F}\) be a filter on \(S\) with \(\overline{ \mathscr{F}}\) a closed subsemigroup of \(\beta S\). If \(A\subseteq S\) such that there exists \(q\in \overline{ \mathscr {F}}\) with \(A\in \mathsf{Syn}(\mathscr{F},\mathsf{Thick}(\mathscr{F},q))\), must \(A\) be a member of \(\mathsf{PS}(\mathscr{F},\mathscr{F})\)?''