An answer to Goswami’s question and new sources of \(IP^{\star }\)-sets containing combined zigzag structure

A set is called an IP-set in a semigroup \(\left( S,\cdot \right) \) $\left(S,,,·\right)$ if it contains finite products of a sequence. A set that intersects all IP-sets is called an \(IP^\star \) $I{P}^{\star }$ -set. It is a well-known and established result by Bergelson and Hindman that if A is an \(IP^{\star }\) $I{P}^{\star }$ -set, then for any sequence \(\langle x_{n}\rangle _{n=1}^{\infty }\) ${⟨x_n⟩}_{n=1}^{\infty }$ in \(\mathbb {N}\) $\mathbb{N}$ (the set of positive integers), there exists a sum subsystem \(\langle y_{n}\rangle _{n=1}^{\infty }\) ${⟨y_n⟩}_{n=1}^{\infty }$ such that \(FS\left( \langle y_{n}\rangle _{n=1}^{\infty }\right) \cup FP \left( \langle y_{n}\rangle _{n=1}^{\infty }\right) \subset A\) $FS\left({⟨y_n⟩}_{n=1}^{\infty }\right)\cup FP\left({⟨y_n⟩}_{n=1}^{\infty }\right)\subset A$ . In [Topology Appl., 300, article no. 107752 (2021)], Goswami posed the question: if we replace the single sequence by l many sequences, then is it possible to obtain a sum subsystem such that all of its zigzag finite sums and products will be in A? Goswami has given affirmative answers only for dynamical \(IP^{\star }\) $I{P}^{\star }$ -sets that are not equivalent to \(IP^{\star }\) $I{P}^{\star }$ -sets but are significantly stronger. We answer Goswami’s question negatively and introduce two notions: k- \(MIP^{\star }\) $MI{P}^{\star }$ -sets and k- \(RIP^{\star }\) $RI{P}^{\star }$ -sets, which will be proved to satisfy the conclusion of Goswami’s question.