Definition 1.4.25 (Lexicographical order).  Let S, T be sets,  be a partial order on S,  be a partial order on T. We define:

This definition constructs a partial order on the Cartesian product of two sets S and T from partial orders on S and T, combining the two partial orders lexicographically. It is used to define multiplication of ordinal numbers (see $../Numbers/Ordinal/product).