Product order explained

\preceq and \sqsubseteq on a set A and B , respectively, the product order^[1] (also called the coordinatewise order^{[2] [3]} or componentwise order^{[1] [4]}) is a partial ordering \leq on the Cartesian product A x B. Given two pairs \left(a _{1,} b _{1\right)} and \left(a _{2,} b _{2\right)} in A x B, declare that \left(a _{1,} b _{1\right)} \leq \left(a _{2,} b _{2\right)} if a _{1} \preceq a _{2} and b _{1} \sqsubseteq b _{2.}

Another possible ordering on

A x B is the lexicographical order. It is a total ordering if both A and B are totally ordered. However the product order of two total orders 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 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.^[5]

The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions.^[4]

The product order generalizes to arbitrary (possibly infinitary) Cartesian products. Suppose

A ≠ \varnothing is a set and for every a \in A, \left(I _{a,} \leq\right) is a preordered set. Then the on \prod _{a} I _{a} is defined by declaring for any i _{\bull} = \left(i _{a\right)} _{a} and j _{\bull} = \left(j _{a\right)} _{a} in \prod _{a} I _{a,} that i _{\bull} \leq j _{\bull} 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} \{ 0 , 1 \} can be identified with the inclusion ordering of subsets of A. ^[6]

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.^[4]

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

Notes and References

1. Book: Sudhir R. Ghorpade. Balmohan V. Limaye. A Course in Multivariable Calculus and Analysis. 2010. Springer. 978-1-4419-1621-1. 5.
2. Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 18
3. Book: Alexander Shen. Nikolai Konstantinovich Vereshchagin. Basic Set Theory. 2002. American Mathematical Soc.. 978-0-8218-2731-4. 43.
4. Book: Paul Taylor. Practical Foundations of Mathematics. 1999. Cambridge University Press. 978-0-521-63107-5. 144–145 and 216.
5. Book: Egbert Harzheim. Ordered Sets. 2006. Springer. 978-0-387-24222-4. 86–88.
6. Book: Victor W. Marek. Introduction to Mathematics of Satisfiability. 2009. CRC Press. 978-1-4398-0174-1. 17.