In mathematics, given partial orders \preceq and \sqsubseteq on sets A and B, respectively, the product order[1][2][3][4] (also called the coordinatewise order[5][3][6] or componentwise order[2][7]) is a partial order \leq 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 a_1 \preceq a_2 and b_1 \sqsubseteq b_2.
Another possible order on A \times B is the lexicographical order. It is a total order 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 order 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.[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 A \neq \varnothing is a set and for every a \in A, \left(I_a, \leq\right) is a preordered set.
Then the on \prod_{a \in A} I_a is defined by declaring for any i_{\bull} = \left(i_a\right)_{a \in A} and j_{\bull} = \left(j_a\right)_{a \in A} in \prod_{a \in A} I_a, that
i_{\bull} \leq j_{\bull}if and only ifi_a \leq j_afor everya \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 order 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]
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
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.