Product order

For the business concept, see purchase order.

In mathematics, given two ordered sets A and B, one can induce a partial ordering on the Cartesian product A × B. Given two pairs (a₁,b₁) and (a₂,b₂) in A × B, one sets (a₁,b₁) ≤ (a₂,b₂) if and only if a₁ ≤ a₂ and b₁ ≤ b₂. This ordering is called the product order, or alternatively the coordinatewise order, or even the componentwise order.

Another possible ordering on A × B is the lexicographical order. Unlike the latter, the product order of two totally ordered sets is not total. For example, the pairs (0, 1) and (1, 0) are incomparable in the product order of 0 < 1 with itself. The lexicographic order of totally ordered sets is however a linear extension of their product order. In general, the product order is a subrelation of the lexicographic order.

The Cartesian product with product order is the categorical product in the category of partially ordered sets with monotone functions.

The product order generalizes to arbitrary (possibly infinitary) Cartesian products. Furthermore, given a set A, the product order over the Cartesian product ∏_A{0, 1} can be identified with the inclusion ordering of subsets of A.

The notion applies equally well to preorders. The product order is also the categorical product in a number of richer categories, including lattices and Boolean algebras.

See also

• 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