# Product order

In mathematics, given two preordered sets ${\displaystyle A}$ and ${\displaystyle B,}$ 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 ${\displaystyle A\times B.}$ Given two pairs ${\displaystyle \left(a_{1},b_{1}\right)}$ and ${\displaystyle \left(a_{2},b_{2}\right)}$ in ${\displaystyle A\times B,}$ declare that ${\displaystyle \left(a_{1},b_{1}\right)\leq \left(a_{2},b_{2}\right)}$ if and only if ${\displaystyle a_{1}\leq a_{2}}$ and ${\displaystyle b_{1}\leq b_{2}.}$

Hasse diagram of the product order on ×ℕ

Another possible ordering on ${\displaystyle A\times B}$ 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 ${\displaystyle (0,1)}$ and ${\displaystyle (1,0)}$are incomparable in the product order of the ordering ${\displaystyle 0<1}$ 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. Suppose ${\displaystyle A\neq \varnothing }$ is a set and for every ${\displaystyle a\in A,}$ ${\displaystyle \left(I_{a},\leq \right)}$ is a preordered set. Then the product preorder on ${\displaystyle \prod _{a\in A}I_{a}}$ is defined by declaring for any ${\displaystyle i_{\bullet }=\left(i_{a}\right)_{a\in A}}$ and ${\displaystyle j_{\bullet }=\left(j_{a}\right)_{a\in A}}$ in ${\displaystyle \prod _{a\in A}I_{a},}$ that

${\displaystyle i_{\bullet }\leq j_{\bullet }}$ if and only if ${\displaystyle i_{a}\leq j_{a}}$ for every ${\displaystyle a\in A.}$

If every ${\displaystyle \left(I_{a},\leq \right)}$ is a partial order then so is the product preorder.

Furthermore, given a set ${\displaystyle A,}$ the product order over the Cartesian product ${\displaystyle \prod _{a\in A}\{0,1\}}$ can be identified with the inclusion ordering of subsets of ${\displaystyle A.}$[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. ^ 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.
3. ^ a b c Egbert Harzheim (2006). Ordered Sets. Springer. pp. 86–88. ISBN 978-0-387-24222-4.
4. ^ a b 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. ^ a b c Paul Taylor (1999). Practical Foundations of Mathematics. Cambridge University Press. pp. 144–145 and 216. ISBN 978-0-521-63107-5.