In mathematics, given two preordered sets and the product order (also called the coordinatewise order or componentwise order) is a partial ordering on the Cartesian product Given two pairs and in declare that if and only if and
Another possible ordering on is the lexicographical order, which is a total ordering. However the product order of two totally ordered sets is not in general total; for example, the pairs and are incomparable in the product order of the ordering with itself. The lexicographic order of totally ordered sets 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 
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