 Let \(I\) be an arbitrary set, which may be empty, finite, or infinite. A family \(\langle A_{i}\rangle_{i\in I}\) of algebras \((X_i, f_{1}^i,\ldots, f_k^i)\) indexed by \(I\) consists of one algebra \(A_i\) for each element \(i\) of \(I\). We define the direct product \(\Pi A_i\) (or \(\Pi_{i\in I} A_i\) in full) of such a family as follows.