ABSTRACT:An IP* set in a semigroup (S,·) is a set which meets every set of the form FP(〈xn〉∞n=1) = {∏nεFxn: F is a finite nonempty subset of ℕ}, where the products are taken in increasing order of indices. We show here, using the Stone‐Cech compactification of the product space S1×S2×…×Sℓ, that if each Si is commutative, then whenever C is an IP* set in S1×S2×…×Sℓ, and for each iε {1, 2, …, ℓ}, 〈xi,n〉∞n=1 is a sequence in Si, C contains Cartesian products of arbitrarily large finite substructures of FP(〈x1,nℓn=1〉) ×FP(〈x2,nℓn=1〉) ×…×FP(〈xℓ,nℓn=1〉). (The notion of “substructure” is made precise in Definition 2.4.) We show further that C need not contain any product of infinite substructures and that the commutativity hypothesis may not be omitted. Similar results apply to arbitrary finite products of semigroups. By way of contrast, we show in Theorem 2.3 that a much stronger conclusion holds for some cell of any finite partition of S1×S2×…×Sℓ, without even any commutativity assumptions.