In mathematics, given partial orders, respectively, the product order (also called the coordinatewise order or componentwise order on the Cartesian product

is the lexicographical order. It is a total order are totally ordered. However the product order of two total orders with itself. The lexicographic combination of two total orders 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.

is a preordered set. Then the that

is a partial order then so is the product preorder.

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

References