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