# Product order

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In mathematics, given two preordered sets $A$ and $B,$ the product order     (also called the coordinatewise order    or componentwise order   ) is a partial ordering on the Cartesian product $A\times B.$ Given two pairs $\left(a_{1},b_{1}\right)$ and $\left(a_{2},b_{2}\right)$ in $A\times B,$ declare that $\left(a_{1},b_{1}\right)\leq \left(a_{2},b_{2}\right)$ if and only if $a_{1}\leq a_{2}$ and $b_{1}\leq b_{2}.$ Another possible ordering on $A\times B$ 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 $(0,1)$ 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. 

The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions. 

The product order generalizes to arbitrary (possibly infinitary) Cartesian products. Suppose $A\neq \varnothing$ is a set and for every $a\in A,$ $\left(I_{a},\leq \right)$ is a preordered set. Then the product preorder on $\prod _{a\in A}I_{a}$ is defined by declaring for any $i_{\bullet }=\left(i_{a}\right)_{a\in A}$ and $j_{\bullet }=\left(j_{a}\right)_{a\in A}$ in $\prod _{a\in A}I_{a},$ that

$i_{\bullet }\leq j_{\bullet }$ if and only if $i_{a}\leq j_{a}$ for every $a\in A.$ If every $\left(I_{a},\leq \right)$ is a partial order then so is the product preorder.

Furthermore, given a set $A,$ the product order over the Cartesian product $\prod _{a\in A}\{0,1\}$ can be identified with the inclusion ordering of subsets of $A.$ 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. 

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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.