Another possible ordering on is the lexicographical order, which is a total ordering. However the product order of two total orders is not in general total; for example, the pairs and are incomparable in the product order of the ordering with itself. The lexicographic combination of two total orders is a linear extension of their product order, and thus the product order is a subrelation of the lexicographic order.
The product order generalizes to arbitrary (possibly infinitary) Cartesian products.
Suppose is a set and for every is a preordered set.
Then the product preorder on is defined by declaring for any and in that
if and only if for every
If every is a partial order then so is the product preorder.
Furthermore, given a set the product order over the Cartesian product can be identified with the inclusion ordering of subsets of