$IP^\star$ set in product space of countable adequate commutative partial semigroups

TL;DR

This paper extends a known combinatorial structure result from product spaces of commutative semigroups to the broader context of countable adequate commutative partial semigroups, revealing rich structural properties.

Contribution

It generalizes a theorem about $IP^{ ext{star}}$ sets from semigroups to partial semigroups, broadening the scope of combinatorial algebra.

Findings

$IP^{ ext{star}}$ sets in partial semigroups contain large structured products.

Extension of Bergelson and Hindman's result to partial semigroups.

Demonstrates rich combinatorial structure in product spaces of partial semigroups.

Abstract

A partial semigroup is a set with restricted binary operation. In this work we will extend a result due to V. Bergelson and N. Hindman concerning the rich structure presented in the product space of semigroups to partial semigroup. An $IP^{\star}$ set in a semigroup is a set that intersect every set of the form $\left\{ FS(x_{n})_{n=1}^{\infty}:x_{n}\in S\right\} $. V. Bergelson and N. Hindman proved that if $S_{1},S_{2},\ldots,S_{l}$ are finite collection of commutative semigroup, then under certain condition, an $IP^{\star}$ set in $S_{1}\times S_{2}\times\ldots\times S_{l}$ contains cartesian products of arbitrarily large finite substructures of the form $FS\left(x_{1,n}\right)_{n=1}^{\infty}\times FS\left(x_{2,n}\right)_{n=1}^{\infty}\times\ldots\times FS\left(x_{l,n}\right)_{n=1}^{\infty}$. In this work we will extend this result to countable adequate commutative partial semigroup.