# Product order

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 a1a2 and b1b2.

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 cartesian product with the 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 $\prod _{A}\{0,1\}$ 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.