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 (a₁, b₁) and (a₂, b₂) in A × B, one defines (a₁, b₁) ≤ (a₂, b₂) if and only if a₁ ≤ a₂ and b₁ ≤ b₂.

For the business concept, see purchase order.

Hasse diagram of the product order on ℕ×ℕ

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 ${\displaystyle \prod _{A}\{0,1\}}$ 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

1. Neggers, J.; Kim, Hee Sik (1998), "4.2 Product Order and Lexicographic Order", Basic Posets, World Scientific, pp. 64–78, ISBN 9789810235895
2. Sudhir R. Ghorpade; Balmohan V. Limaye (2010). A Course in Multivariable Calculus and Analysis. Springer. p. 5. ISBN 978-1-4419-1621-1.
3. Egbert Harzheim (2006). Ordered Sets. Springer. pp. 86–88. ISBN 978-0-387-24222-4.
4. Victor W. Marek (2009). Introduction to Mathematics of Satisfiability. CRC Press. p. 17. ISBN 978-1-4398-0174-1.
5. Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 18
6. Alexander Shen; Nikolai Konstantinovich Vereshchagin (2002). Basic Set Theory. American Mathematical Soc. p. 43. ISBN 978-0-8218-2731-4.
7. 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
• Ordered vector space