{{Short description|Construction in order theory}} {{for|the business concept|purchase order}} [[File:N-Quadrat, gedreht.svg|thumb|300px|Hasse diagram of the product order on <math>\mathbb{N}</math>×<math>\mathbb{N}</math>]] In mathematics, given partial orders <math>\preceq</math> and <math>\sqsubseteq</math> on sets <math>A</math> and <math>B</math>, respectively, the '''product order'''<ref>{{citation|last1=Neggers|first1=J.|last2=Kim|first2=Hee Sik|contribution = 4.2 Product Order and Lexicographic Order|isbn = 9789810235895| pages = 64–78| publisher=World Scientific|title=Basic Posets|url=https://books.google.com/books?id=-ip3-wejeR8C&pg=PA64| year = 1998}}</ref><ref name="anal">{{cite book|author1=Sudhir R. Ghorpade|author2=Balmohan V. Limaye|title=A Course in Multivariable Calculus and Analysis|year=2010|publisher=Springer|isbn=978-1-4419-1621-1|page=5}}</ref><ref name="Harzheim"/><ref name="Marek"/> (also called the '''coordinatewise order'''<ref>Davey & Priestley, ''Introduction to Lattices and Order'' (Second Edition), 2002, p.&nbsp;18</ref><ref name="Harzheim"/><ref>{{cite book|author1=Alexander Shen|author2=Nikolai Konstantinovich Vereshchagin|title=Basic Set Theory|year=2002|publisher=American Mathematical Soc.|isbn=978-0-8218-2731-4|page=43}}</ref> or '''componentwise order'''<ref name="anal"/><ref name="taylor">{{cite book|author=Paul Taylor|title=Practical Foundations of Mathematics|year=1999|publisher=Cambridge University Press|isbn=978-0-521-63107-5|pages=144–145 and 216}}</ref>) is a partial order <math>\leq</math> on the Cartesian product <math>A \times B.</math> Given two pairs <math>\left(a_1, b_1\right)</math> and <math>\left(a_2, b_2\right)</math> in <math>A \times B,</math> declare that <math>\left(a_1, b_1\right) \leq \left(a_2, b_2\right)</math> if <math>a_1 \preceq a_2</math> and <math>b_1 \sqsubseteq b_2.</math>

Another possible order on <math>A \times B</math> is the lexicographical order. It is a total order if both <math>A</math> and <math>B</math> are totally ordered. However the product order of two total orders is not in general total; for example, the pairs <math>(0, 1)</math> and <math>(1, 0)</math> are incomparable in the product order of the order <math>0 < 1</math> 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.<ref name="Harzheim">{{cite book|author=Egbert Harzheim|title=Ordered Sets|year=2006|publisher=Springer|isbn=978-0-387-24222-4|pages=86–88}}</ref>

The Cartesian product with the product order is the categorical product in the category of partially ordered sets with monotone functions.<ref name="taylor"/>

The product order generalizes to arbitrary (possibly infinitary) Cartesian products. Suppose <math>A \neq \varnothing</math> is a set and for every <math>a \in A,</math> <math>\left(I_a, \leq\right)</math> is a preordered set. Then the {{em|{{visible anchor|product preorder}}}} on <math>\prod_{a \in A} I_a</math> is defined by declaring for any <math>i_{\bull} = \left(i_a\right)_{a \in A}</math> and <math>j_{\bull} = \left(j_a\right)_{a \in A}</math> in <math>\prod_{a \in A} I_a,</math> that

:<math>i_{\bull} \leq j_{\bull}</math> if and only if <math>i_a \leq j_a</math> for every <math>a \in A.</math>

If every <math>\left(I_a, \leq\right)</math> is a partial order then so is the product preorder.

Furthermore, given a set <math>A,</math> the product order over the Cartesian product <math>\prod_{a \in A} \{0, 1\}</math> can be identified with the inclusion order of subsets of <math>A.</math><ref name="Marek">{{cite book|author=Victor W. Marek|title=Introduction to Mathematics of Satisfiability|year=2009|publisher=CRC Press|isbn=978-1-4398-0174-1|page=17}}</ref>

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.<ref name="taylor"/>

==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 * {{annotated link|Ordered vector space}}

== References == {{reflist}}

{{Order theory}}

Category:Order theory

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