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In mathematics, given a partial order and on a set and , respectively, the **product order**^{[1]}^{[2]}^{[3]}^{[4]} (also called the **coordinatewise order**^{[5]}^{[3]}^{[6]} or **componentwise order**^{[2]}^{[7]}) is a partial ordering on the Cartesian product Given two pairs and in declare that if and

Another possible ordering on is the lexicographical order. It is a total ordering if both and are totally ordered. However the product order of two total orders is not in general total; for example, the pairs and are incomparable in the product order of the ordering 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.^{[3]}

The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions.^{[7]}

The product order generalizes to arbitrary (possibly infinitary) Cartesian products.
Suppose is a set and for every is a preordered set.
Then the *product preorder* on is defined by declaring for any and in that

- if and only if for every

If every is a partial order then so is the product preorder.

Furthermore, given a set the product order over the Cartesian product can be identified with the inclusion ordering of subsets of ^{[4]}

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.^{[7]}

- 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
- Ordered vector space – Vector space with a partial order

**^**Neggers, J.; Kim, Hee Sik (1998), "4.2 Product Order and Lexicographic Order",*Basic Posets*, World Scientific, pp. 64–78, ISBN 9789810235895- ^
^{a}^{b}Sudhir R. Ghorpade; Balmohan V. Limaye (2010).*A Course in Multivariable Calculus and Analysis*. Springer. p. 5. ISBN 978-1-4419-1621-1. - ^
^{a}^{b}^{c}Egbert Harzheim (2006).*Ordered Sets*. Springer. pp. 86–88. ISBN 978-0-387-24222-4. - ^
^{a}^{b}Victor W. Marek (2009).*Introduction to Mathematics of Satisfiability*. CRC Press. p. 17. ISBN 978-1-4398-0174-1. **^**Davey & Priestley,*Introduction to Lattices and Order*(Second Edition), 2002, p. 18**^**Alexander Shen; Nikolai Konstantinovich Vereshchagin (2002).*Basic Set Theory*. American Mathematical Soc. p. 43. ISBN 978-0-8218-2731-4.- ^
^{a}^{b}^{c}Paul Taylor (1999).*Practical Foundations of Mathematics*. Cambridge University Press. pp. 144–145 and 216. ISBN 978-0-521-63107-5.