Definition 5.6. Let ${(A_{j})}_{j \in J}$ , $U$ (ultrafilter on $J$ ), $\sim$ as before, and let $n \in ℕ$ .

For each $j \in J$ , suppose $B_{j} \subseteq A_{j}^{n}$ . Define

\begin{array}{l} [{(B_{j})}_{j \in J}] & = {([{(a_{j}^{1})}_{j \in J}], \dots, [{(a_{j}^{n})}_{j \in J}]) \in {(\prod_{j \in J} A_{j} ∕ U)}^{n} : {j \in J : (a_{j}^{1}, \dots, a_{j}^{n}) \in B_{j}} \in U} \\ \subseteq {(\prod_{j \in J} A_{j} ∕ U)}^{n} \end{array}