In mathematics, given two partially ordered sets A and B, the product order (also called the coordinatewise order or componentwise order) is a partial ordering on the cartesian product A × B. Given two pairs (a1, b1) and (a2, b2) in A × B, one defines (a1, b1) ≤ (a2, b2) if and only if a1 ≤ a2 and b1 ≤ b2.
Another possible ordering on A × B 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 (0, 1) and (1, 0) are incomparable in the product order of the ordering 0 < 1 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. Furthermore, given a set A, the product order over the cartesian product can be identified with the inclusion ordering of subsets of A.
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- direct product of binary relations
- examples of partial orders
- Star product, a different way of combining partial orders
- orders on the cartesian product of totally ordered sets
- Ordinal sum of partial orders
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