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A partial semigroup is a set with restricted binary operation. In this workwe will extend a result due to V. Bergelson and N. Hindman concerning the richstructure 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 ofcommutative semigroup, then under certain condition, an $IP^{\star}$ set in$S_{1}\times S_{2}\times\ldots\times S_{l}$ contains cartesian products ofarbitrarily large finite substructures of the form$FS\left(x_{1,n}\right)_{n=1}^{\infty}\timesFS\left(x_{2,n}\right)_{n=1}^{\infty}\times\ldots\timesFS\left(x_{l,n}\right)_{n=1}^{\infty}$. In this work we will extend this resultto countable adequate commutative partial semigroup.

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