Product order
In mathematics, given a partial order and on a set and , respectively, 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 Given two pairs and in declare that if and
Another possible ordering on is the lexicographical order. It is a total ordering if both and are totally ordered. However the product order of two total orders is not in general total; for example, the pairs and are incomparable in the product order of the ordering 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 is a set and for every is a preordered set. Then the product preorder on is defined by declaring for any and in that
- if and only if for every
If every is a partial order then so is the product preorder.
Furthermore, given a set the product order over the Cartesian product can be identified with the inclusion ordering of subsets of [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
[edit]- 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 – Vector space with a partial order
References
[edit]- ^ Neggers, J.; Kim, Hee Sik (1998), "4.2 Product Order and Lexicographic Order", Basic Posets, World Scientific, pp. 64–78, ISBN 9789810235895
- ^ 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.
- ^ a b c Egbert Harzheim (2006). Ordered Sets. Springer. pp. 86–88. ISBN 978-0-387-24222-4.
- ^ a b 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.
- ^ a b c Paul Taylor (1999). Practical Foundations of Mathematics. Cambridge University Press. pp. 144–145 and 216. ISBN 978-0-521-63107-5.