# Product order

In mathematics, given two partially ordered sets *A* and *B*, 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 *A* × *B*. Given two pairs (*a*_{1}, *b*_{1}) and (*a*_{2}, *b*_{2}) in *A* × *B*, one defines (*a*_{1}, *b*_{1}) ≤ (*a*_{2}, *b*_{2}) if and only if *a*_{1} ≤ *a*_{2} and *b*_{1} ≤ *b*_{2}.

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.[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. Furthermore, given a set *A*, the product order over the cartesian product
can be identified with the inclusion ordering of subsets of *A*.[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]

## References

- Neggers, J.; Kim, Hee Sik (1998), "4.2 Product Order and Lexicographic Order",
*Basic Posets*, World Scientific, pp. 64–78, ISBN 9789810235895 - Sudhir R. Ghorpade; Balmohan V. Limaye (2010).
*A Course in Multivariable Calculus and Analysis*. Springer. p. 5. ISBN 978-1-4419-1621-1. - Egbert Harzheim (2006).
*Ordered Sets*. Springer. pp. 86–88. ISBN 978-0-387-24222-4. - 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. - Paul Taylor (1999).
*Practical Foundations of Mathematics*. Cambridge University Press. pp. 144–145 and 216. ISBN 978-0-521-63107-5.

## 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