Zigzag structures in IP⁎ sets and dynamical IP⁎ sets

Abstract

A set is called an IP set in a semigroup (S,⋅) if it contains all finite products of a sequence. A set which intersects with all IP sets is known as IP⁎ set. V. Bergelson and N. Hindman proved if A is an IP⁎ set in (N,+), then for any sequence 〈xn〉n=1∞, there exists a sum subsystem 〈yn〉n=1∞ such that both FS(〈yn〉n=1∞) and FP(〈yn〉n=1∞) are contained in A. S. Goswami asked if we replace the single sequence by l-sequence, then is it possible to obtain a sum subsystem such that all of its zigzag finite sums and products will be in A. He proved that for certain IP⁎ sets (known as dynamical IP⁎ sets) this is possible. In this article, we will show that if A is an IP⁎ set, then for certain l-sequence, there exists a diagonal sum subsystem such that all of its zigzag finite sums and products will be in A.