Product 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. Another possible ordering on A × B is the lexicographical order.

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

See also

• direct product of binary relations
• examples of partial orders
• orders on the Cartesian product of totally ordered sets